Integer Sequences Connected with Extensions of the Bell Polynomials

Pierpaolo Natalini Università degli Studi Roma Tre Dipartimento di Matematica e Fisica Largo San Leonardo Murialdo, 1 00146 - Roma Italy

               natalini@mat.uniroma3.it

Paolo Emilio Ricci International Telematic University UniNettuno Corso Vittorio Emanuele II, 39 00186 - Roma Italy

              paoloemilioricci@gmail.com

Abstract

The Encyclopedia of Integer Sequences includes some sequences that are connected with the Bell numbers and that have a particular combinatorial meaning. In this article, we find a general meaning for framing sequences, including the above mentioned ones. Furthermore, by using Laguerre-type derivatives, we derive the Laguerre-type Bell numbers of higher order, showing, as a by-product, that it is possible to construct new integer sequences which are not included in the Encyclopedia.

1 Introduction

The Bell polynomials [3] are a mathematical tool for representing the nth derivative of a composite function. They are strictly related to partitions [1, 2, 24]. They are also applied in many different frameworks such as the Blissard problem [24, p. 46], the representation of Lucas polynomials of the first and second kind [10], the construction of representation formulas for the Newton sum rules for the zeros of polynomials [16, 17], the recurrence relations for a class of Freud-type polynomials [6], the representation formulas for the symmetric functions of a countable set of numbers generalizing the classical algebraic Newton-Girard formulas [8]. In particular, Cassisa and Ricci [8] found reduction formulas for the orthogonal invariants of a strictly positive compact operator, obtaining, as a by-product, the so-called Robert formulas [25].

Recently Kataria and Vellaisamy [14, 15] obtained connections of Bell polynomials with another important class of polynomials known as Adomian polynomials.

Some generalized forms of Bell polynomials can be found in the literature, (see e.g., [7, 12, 18, 19, 21, 22, 23]). For instance, some papers [7, 22] are concerned with the multidimensional case.

The authors [18] introduced the higher order Bell polynomials and their main properties and they [21] used some recursion formulas to compute, by means of the program Mathematica^{\begin{matrix}c \\ ◯ \end{matrix}}, the rth order complete Bell polynomials B_{n}^{[r]} (for r = 2, 3, 4, 5) and the relevant Bell numbers b_{n}^{[r]}.

In this article, after briefly recalling this theory, we show how to merge a family of integer sequences, already known in the Encyclopedia of Integer Sequences [28], into a general framework.

Furthermore, we construct new sequences of integer numbers, not included in the Encyclopedia, related to the Laguerre-type Bell polynomials. To this aim, we recall, in Section 6, a short introduction to the Laguerre derivative.

2 Recalling the Bell polynomials

By considering the composite function Φ(t) := f(g(t)) of functions x = g(t) and y = f(x), defined in suitable intervals of the real axis and n times differentiable with respect to the relevant independent variables and by using the notation

Φ_{h} := D_{t}^{h}Φ(t),  f_{h} := D_{x}^{h}f(x)|_{x=g(t)},  g_{h} := D_{t}^{h}g(t) ,  (1)

and

([f, g]_{n}) := (f_{1}, g_{1}; f_{2}, g_{2}; …; f_{n}, g_{n}) ,  (2)

the Bell polynomials are defined as follows:

Y_{n} ([f, g]_{n}) := Φ_{n}.  (3)

Inductively, by letting

[g]_{n} := (g_{1}, g_{2}, …, g_{n}),

we can write

Y_{n} ([f, g]_{n}) = ∑_{k=1}^{n} A_{n,k}([g]_{n})f_{k},  (4)

where the coefficient A_{n,k}, for any k = 1, …, n, is a polynomial in g_{1}, g_{2}, …, g_{n}, homogeneous of degree k and isobaric of weight n (i.e., it is a linear combination of monomials g_{1}^{k_{1}}g_{2}^{k_{2}} ⋯ g_{n}^{k_{n}} whose weight is constantly given by k_{1} + 2k_{2} + … + nk_{n} = n).

Proposition 1. The Bell polynomials satisfy the recurrence relation

\{ \begin{matrix}Y_{0} ([f, g]_{0}) := f_{1} \\ Y_{n+1} ([f, g]_{n+1}) = ∑_{k=0}^{n} \binom{n}{k}Y_{n−k} ([f_{1}, g]_{n−k}) g_{k+1}, \end{matrix}  (5)

where

([f_{1}, g]_{n−k}) := (f_{2}, g_{1}; f_{3}, g_{2}; …; f_{n−k+1}, g_{n−k}).

The Faà di Bruno formula [11, 26, 27] gives an explicit expression for the Bell polynomials, however, as this formula makes use of partitions, it is not useful in practice for computing higher order Bell polynomials, whereas this can be done in a easy way by means of the following recursion formula for the coefficients A_{n,k} in equation (4) which are known as partial Bell polynomials.

Theorem 2. We have, for all integers n,

A_{n+1,1} = g_{n+1},  A_{n+1,n+1} = g_{1}^{n+1}.  (6)

Furthermore, for all k = 1, 2, …, n − 1, the A_{n,k} coefficients can be computed by the recurrence relation

A_{n+1,k+1}([g]_{n+1}) = ∑_{h=0}^{n−k} \binom{n}{h}A_{n−h,k}([g]_{n−h})g_{h+1}.  (7)

Definition 3. The complete Bell polynomials are defined by

B_{n}([g]_{n}) = Y_{n}(1, g_{1}; 1, g_{2}; … 1, g_{n}) = ∑_{k=1}^{n} A_{n,k}([g]_{n}),  (8)

and the Bell numbers by

b_{n} = Y_{n}(1, 1; 1, 1; …; 1, 1) = ∑_{k=1}^{n} A_{n,k}(1, 1, …, 1).  (9)

3 Bell polynomials of order r

Bernardini et al. [7] introduced the multi-dimensional Bell polynomials of higher order. Here, we briefly recall this extension of the classical Bell polynomials only in the one-dimensional case.

Consider Φ(t) := f(φ^{(1)}(φ^{(2)}(⋯ (φ^{(r)}(t))))), i.e., the composition of functions x^{(r)} = φ^{(r)}(t), …, x^{(2)} = φ^{(2)} (x^{(3)}), x^{(1)} = φ^{(1)} (x^{(2)}), y = f (x^{(1)}) defined in suitable intervals of the real axis, and suppose that the functions φ^{(r)}, …, φ^{(2)}, φ^{(1)}, f are n times differentiable with respect to the relevant independent variables so that, by using the chain rule, Φ(t) can be differentiated n times with respect to t.

We use the following notation

Φ_{h} := D_{t}^{h}Φ(t),
f_{h} := D_{x^{(1)}}^{h}f|_{x^{(1)}=φ^{(1)}(···(φ^{(r)}(t)))},
φ_{h}^{(1)} := D_{x^{(2)}}^{h}φ^{(1)}|_{x^{(2)}=φ^{(2)}(···(φ^{(r)}(t)))},  (10)
⋮
φ_{h}^{(r)} := D_{t}^{h}φ^{(r)}(t),

and

([f, φ^{(1)}, …, φ^{(r)}]_{n}) := (f_{1}, φ_{1}^{(1)}, …, φ_{1}^{(r)}; …; f_{n}, φ_{n}^{(1)}, …, φ_{n}^{(r)}).

Then the nth derivative of the function Φ allows us to define the one-dimensional Bell polynomials of order r, Y_{n}^{[r]}, as follows:

Y_{n}^{[r]} ([f, φ^{(1)}, …, φ^{(r)}]_{n}) := Φ_{n}.  (11)

For r = 1 we obtain the ordinary Bell polynomials Y_{n}^{[1]} ([f, φ^{(1)}]_{n}) = Y_{n} ([f, φ^{(1)}]_{n}). The first polynomials have the following explicit expressions

Y_{1}^{[r]} ([f, φ^{(1)}, …, φ^{(r)}]_{1}) = f_{1}φ_{1}^{(1)} ⋯ φ_{1}^{(r)}  (12)
Y_{2}^{[r]} ([f, φ^{(1)}, …, φ^{(r)}]_{2}) = f_{2}(φ_{1}^{(1)} ⋯ φ_{1}^{(r)})^{2} + f_{1}φ_{2}^{(1)}(φ_{1}^{(2)} ⋯ φ_{1}^{(r)})^{2}
+f_{1}φ_{1}^{(1)}φ_{2}^{(2)}(φ_{1}^{(3)} ⋯ φ_{1}^{(r)})^{2} + f_{1}φ_{1}^{(1)}φ_{1}^{(2)} ⋯ φ_{1}^{(r−1)}φ_{2}^{(r)}.

In general, we have

Y_{n}^{[r]}([f, φ^{(1)}, …, φ^{(r)}]_{n}) = ∑_{k=1}^{n} A_{n,k}^{[r]}([φ^{(1)}, …, φ^{(r)}]_{n})f_{k}.  (13)

The authors [18] proved the following useful properties for the polynomials Y_{n}^{[r]} Theorem 4. For every integer n, the polynomials Y_{n}^{[r]} are expressed in terms of the Bell polynomials of lower order, by means of the following equation

Y_{n}^{[r]} ([f, φ^{(1)}, …, φ^{(r)}]_{n}) = Y_{n} ([f, Y^{[r−1]} ([φ^{(1)}, …, φ^{(r)}])]_{n}) ,  (14)

where

([f, Y^{[r−1]} ([φ^{(1)}, …, φ^{(r)}])]_{n})
:= (f_{1}, Y_{1}^{[r−1]} ([φ^{(1)}, …, φ^{(r)}]_{1}); …; f_{n}, Y_{n}^{[r−1]} ([φ^{(1)}, …, φ^{(r)}]_{n})).

Theorem 5. The following recurrence relation for the Bell polynomials Y_{n}^{[r]} holds

\{ \begin{matrix}Y_{0}^{[r]} ([f, φ^{(1)}, …, φ^{(r)}]_{0}) = f_{1} \\ Y_{n+1}^{[r]} ([f, φ^{(1)}, …, φ^{(r)}]_{n+1}) = ∑_{k=0}^{n} \binom{n}{k}Y_{n−k}^{[r]} ([f_{1}, φ^{(1)}, …, φ^{(r)}]_{n−k}) \\ × Y_{k+1}^{[r−1]} ([φ^{(1)}, …, φ^{(r)}]_{k+1}) , \end{matrix}  (15)

where

([f_{1}, φ^{(1)}, …, φ^{(r)}]_{n−k}) := (f_{2}, φ_{1}^{(1)}, …, φ_{1}^{(r)}; …; f_{n−k+1}, φ_{n−k}^{(1)}, …, φ_{n−k}^{(r)}) .

The recurrence relation (6)–(7) can be generalized as follows:

Theorem 6. For all integer n, we have

\begin{matrix}A_{n+1,1}^{[r]} = Y_{n+1}^{[r−1]} ([φ^{(1)}, …, φ^{(r)}]_{n+1}) , \\ A_{n+1,n+1}^{[r]} = (Y_{1}^{[r−1]} ([φ^{(1)}, …, φ^{(r)}]_{1}))^{n+1} = (φ_{1}^{(1)} ⋯ φ_{1}^{(r)})^{n+1} . \end{matrix}  (16)

Furthermore, for all k = 1, 2, …, n − 1, the rth order partial Bell polynomials A_{n,k}^{[r]} satisfy the recursion

\begin{matrix}A_{n+1,k+1}^{[r]} ([φ^{(1)}, …, φ^{(r)}]_{n+1}) = ∑_{h=0}^{n−k} \binom{n}{h}A_{n−h,k}^{[r]} ([φ^{(1)}, …, φ^{(r)}]_{n−h}) \\ × Y_{h+1}^{[r−1]} ([φ^{(1)}, …, φ^{(r)}]_{h+1}) . \end{matrix}  (17)

Definition 7. The complete Bell polynomials of order r, B_{n}^{[r]}, are defined by the equation

\begin{matrix}B_{n}^{[r]} ([φ^{(1)}, …, φ^{(r)}]_{n}) = Y_{n}^{[r]}(1, φ_{1}^{(1)}, …, φ_{1}^{(r)}; …; 1, φ_{n}^{(1)}, …, φ_{n}^{(r)}) \\ = ∑_{k=1}^{n} A_{n,k}^{[r]} ([φ^{(1)}, …, φ^{(r)}]_{n}) , \end{matrix}

and the rth order Bell numbers by

b_{n}^{[r]} = Y_{n}^{[r]}(1, 1, 1; …; 1, 1, 1) = ∑_{k=1}^{n} A_{n,k}^{[r]}(1, 1; …; 1, 1).

4 Higher order Bell numbers, for r = 2, 3, 4, 5

The sequences of higher order Bell numbers, which are presented here, appear in the Encyclopedia of Integer Sequences [28] under \underline{A144150}, arising from a problem of combinatorial analysis and even [4, 13] as the McLaurin coefficients of the functions

exp(exp(exp(x) − 1) − 1),

exp(exp(exp(exp(x) − 1) − 1) − 1),

exp(exp(exp(exp(exp(x) − 1) − 1) − 1) − 1),

exp(exp(exp(exp(exp(exp(x) − 1) − 1) − 1) − 1) − 1),

for the cases r = 2, r = 3, r = 4, r = 5, respectively, and so on for the subsequent values of r. In our approach they assume a more general meaning, as they are independent of the functions f, φ^{(1)}, …, φ^{(r)}.

By using the recurrence relation (16)–(17) and by means of the computer algebra program Mathematica^{\begin{matrix}c \\ ◯ \end{matrix}}, we find the following sequences for the higher order Bell numbers b_{n}^{[2]}, b_{n}^{[3]}, b_{n}^{[4]}, b_{n}^{[5]}, (n = 1, 2, …, 10):

\begin{matrix}n & b_{n}^{[2]} & b_{n}^{[3]} & b_{n}^{[4]} & b_{n}^{[5]} \\ 1 & 1 & 1 & 1 & 1 \\ 2 & 3 & 4 & 5 & 6 \\ 3 & 12 & 22 & 35 & 51 \\ 4 & 60 & 154 & 315 & 561 \\ 5 & 358 & 1304 & 3455 & 7556 \\ 6 & 2471 & 12915 & 44590 & 120196 \\ 7 & 19302 & 146115 & 660665 & 2201856 \\ 8 & 167894 & 1855570 & 11035095 & 45592666 \\ 9 & 1606137 & 26097835 & 204904830 & 1051951026 \\ 10 & 16733779 & 402215465 & 4183174520 & 26740775306 \end{matrix}

Table 1: Higher order Bell numbers for n = 1, 2, …, 10.

The above table can be extended up to the desired order, as the Mathematica^{\begin{matrix}c \\ ◯ \end{matrix}} program runs efficiently.

5 Sequences of integer numbers connected with the above schemes

Putting, for all integers n, b_{n}^{[0]} := 1, b_{n}^{[1]} := b_{n}, and reading Table 1 by row, we find the new array

k = 0 k = 1 k = 2 k = 3 k = 4 k = 5
b_{1}^{[k]}  1  1  1  1  1  1
b_{2}^{[k]}  1  2  3  4  5  6
b_{3}^{[k]}  1  5  12  22  35  51
b_{4}^{[k]}  1  15  60  154 315  561
b_{5}^{[k]}  1  52  358 1304 3455 7556
b_{6}^{[k]}  1  203 2471 12915 44590 120196

Table 2: The above sequences to be read by row.

The construction law of the sequence (b_{3}^{[k]})_{k≥0} (third row) is

b_{3}^{[1]} = b_{2}^{[1]} + b_{2}^{[2]}
b_{3}^{[2]} = b_{2}^{[2]} + b_{2}^{[3]} + b_{2}^{[4]}
⋮  (18)
b_{3}^{[k]} = b_{2}^{[k]} + b_{2}^{[k+1]} + ⋯ + b_{2}^{[2k]}.

Deriving a similar law for the subsequent rows is an open problem, since there is no connection between the meaning of the sequences (b_{3}^{[k]})_{k≥0} and (b_{4}^{[k]})_{k≥0}. In the Encyclopedia

of Integer Sequences, the third row (1, 5, 12, 22, 35, 51, …) appears under \underline{A000326}, and b_{3}^{[n]} is identified, according to the above recursion (18), as the sum of n integers starting from n, or as the pentagonal number n(3n − 1)/2.

The fourth row (1, 15, 60, 154, 315, 561, …) appears under \underline{A005945} and b_{4}^{[n]} is the number of n-step mappings with 4 inputs.

The subsequent rows are not included explicitly in the Encyclopedia, so our approach is useful in order to merge the above sequences into a general framework.

6 Laguerre-type derivatives

Dattoli and Ricci [9] introduced the Laguerre-type derivatives in connection with a differential isomorphism denoted by the symbol T := T_{x}, acting onto the space A := A_{x} of analytic functions of the x variable by means of the correspondence

D := \frac{d}{dx} → D̂_{L} := DxD;  x· → D̂_{x}^{−1},  (19)

where

D̂_{x}^{−1}f(x) := ∫_{0}^{x} f(ξ)dξ,

D̂_{x}^{−n}f(x) := \frac{1}{(n − 1)!} ∫_{0}^{x}(x − ξ)^{n−1}f(ξ)dξ,

and D̂_{x}^{−n} denotes the n-fold R-L integration, so that

T_{x}(x^{n}) = D̂_{x}^{−n}(1) := \frac{1}{(n − 1)!} ∫_{0}^{x}(x − ξ)^{n−1}dξ = \frac{x^{n}}{n!} .  (20)

According to the isomorphism T_{x}, the exponential operator e^{x} is transformed into the first Laguerre-type exponential e_{1}(x) := ∑_{k=0}^{∞} x^{k}/(k!)^{2} which is an eigenfunction of the Laguerre derivative operator D_{L} := DxD. We have, in fact,

T_{x}(e^{x}) = ∑_{k=0}^{∞} \frac{T_{x}(x^{k})}{k!} = ∑_{k=0}^{∞} \frac{x^{k}}{(k!)^{2}} = e_{1}(x),

D̂_{L} e_{1}(ax) = ae_{1}(ax), ∀a ∈ C.

This result can be generalized by considering the rth Laguerre-type exponential e_{r}(x) := ∑_{k=0}^{∞} x^{k}/(k!)^{r+1}, the rth Laguerre-type derivative operator D_{rL} := DxDxD ⋯ DxD (containing r + 1 ordinary derivatives), and the iterated isomorphism T^{r}, since

T_{x}^{r}(e^{x}) = ∑_{k=0}^{∞} \frac{T_{x}(x^{k})}{(k!)^{r}} = ∑_{k=0}^{∞} \frac{x^{k}}{(k!)^{r+1}} = e_{r}(x),

D̂_{rL} e_{r}(ax) = ae_{r}(ax), ∀a ∈ C.

Remark 8. The above results show that, for every positive integer r, we can define a Laguerretype exponential function e_{r}(x), satisfying an eigenfunction property, which is an analog of the elementary property of the exponential. The Laguerre-type exponential function reduces to the classical exponential function when r = 0, so that we can put by definition

e_{0}(x) := e^{x},  D̂_{0L} := D.

Obviously, D̂_{1L} := D̂_{L}.

For this reason we will refer to such functions as L-exponential functions, or shortly L-exponentials.

Bernardini et al. [5] found several applications to the theory of L-exponential functions.

7 Laguerre-type Bell polynomials and numbers

The problem of constructing Bell polynomials can be extended in the natural way to the case of the Laguerre-type derivatives.

By using the notation (1)–(2), we introduce the following definition

Definition 9. The nth Laguerre-type Bell polynomial of order r, denoted by_{rL}Y_{n} (x; [f, g]_{n}), represents the nth Laguerre-type derivative of order r of the composite function f(g(t)).

The authors [19] showed that _{rL}Y_{n} can be expressed as a polynomial in the independent variable x, depending on f_{1}, g_{1}; f_{2}, g_{2}; …; f_{n}, g_{n} in terms of the classical Bell polynomials.

According to a general result due to Viskov [29], the Laguerre derivative satisfy

(D_{L})^{n} = (DxD)^{n} = D^{n}x^{n}D^{n},  (21)

and furthermore, for any order r, it turns out that

(D_{rL})^{n} = (DxDx ⋯ DxD)^{n} = D^{n}x^{n}D^{n}x^{n} ⋯ D^{n}x^{n}D^{n}.  (22)

Therefore, the authors [19] proved the following representation formula for the Laguerretype Bell polynomials of the first order, denoted by _{L}Y_{n},

Theorem 10. The _{L}Y_{n} polynomials are expressed in terms of the ordinary Bell polynomials according to the equation

LY_{n} (x; [f, g]_{n}) = ∑_{k_{=}0}^{n} \frac{n!}{k!}\binom{n}{k}x^{k} Y_{n+k} ([f, g]_{n+k}) .  (23)

Definition 11. The 1st order Laguerre-type Bell numbers are defined by

Lb_{n} :=_{L} Y_{n} (1; 1, 1, …, 1) = ∑_{k_{=}0}^{n} \frac{n!}{k!}\binom{n}{k}Y_{n+k} (1, 1, …, 1) .

The above results can be easily generalized, since

\begin{matrix}(D_{2L})^{n} = (DxDxD)^{n} = D^{n}x^{n} (D^{n}x^{n}D^{n}) = \\ = ∑_{k_{1}=0}^{n} ∑_{k_{2}=0}^{n} \frac{n!}{k_{1}!} \frac{(n + k_{1})!}{(k_{1} + k_{2})!}\binom{n}{k_{1}}\binom{n}{k_{2}}x^{k_{1}+k_{2}}D^{n+k_{1}+k_{2}}, \end{matrix}  (24)

so that the last definition becomes

Definition 12. The 2nd order Laguerre-type Bell numbers are defined by

2Lb_{n} :=_{2L} Y_{n} (1; 1, 1, …, 1) = ∑_{k_{1}=0}^{n} ∑_{k_{2}=0}^{n} \frac{n!}{k_{1}!} \frac{(n + k_{1})!}{(k_{1} + k_{2})!}\binom{n}{k_{1}}\binom{n}{k_{2}}Y_{n+k_{1}+k_{2}} (1, 1, …, 1)

and in general, for every integer r, Definition 13. The rth order Laguerre-type Bell numbers are defined by

\begin{matrix}_{rL}b_{n} :=_{rL} Y_{n} (1; 1, 1, …, 1) = \\ = ∑_{k_{1}=0}^{n} ∑_{k_{2}=0}^{n} ⋯ ∑_{k_{r}=0}^{n} \frac{n!}{k_{1}!} \frac{(n + k_{1})!}{(k_{1} + k_{2})!} ⋯ \frac{(n + k_{1} + k_{2} + ⋯ + k_{r−1})!}{(k_{1} + k_{2} + ⋯ + k_{r})!} \\ × \binom{n}{k_{1}}\binom{n}{k_{2}} ⋯ \binom{n}{k_{r}}Y_{n+k_{1}+k_{2}+···+k_{r}} (1, 1, …, 1) . \end{matrix}  (25)

In particular, for r = 1, r = 2 and r = 3, we obtain respectively

Lb_{n} = ∑_{k=0}^{n} \frac{(n!)^{2}}{(k!)^{2}(n − k)!}b_{n+k},

2Lb_{n} = ∑_{k_{1}=0}^{n} ∑_{k_{2}=0}^{n} \frac{(n!)^{3}(n + k_{1})!}{(k_{1}!)^{2}(k_{2})!(k_{1} + k_{2})!(n − k_{1})!(n − k_{2})!}b_{n+k_{1}+k_{2}},

and

\begin{matrix}_{3L}b_{n} = ∑_{k_{1}=0}^{n} ∑_{k_{2}=0}^{n} ∑_{k_{3}=0}^{n} \\ \frac{(n!)^{4}(n + k_{1})!(n + k_{1} + k_{2})!}{(k_{1}!)^{2}(k_{2})!(k_{1} + k_{2}!)(k_{3})!(k_{1} + k_{2} + k_{3})!(n − k_{1})!(n − k_{2})!(n − k_{3})!}b_{n+k_{1}+k_{2}+k_{3}}. \end{matrix}

According to the above definitions, we compute the following integer sequences relevant to the Laguerre-type Bell numbers of increasing order:

Lb_{1} = b_{1} + b_{2} = 3 Lb_{2} = 2b_{2} + 4b_{3} + b_{4} = 39 Lb_{3} = 6b_{3} + 18b_{4} + 9b_{5} + b_{6} = 971 Lb_{4} = 24b_{4} + 96b_{5} + 72b_{6} + 16b_{7} + b_{8} = 38140 Lb_{5} = 120b_{5} + 600b_{6} + 600b_{7} + 200b_{8} + 25b_{9} + b_{10} = 2126890 Lb_{6} = 720b_{6} + 4320b_{7} + 5400b_{8} + 2400b_{9} + 450b_{10} + 36b_{11} + b_{12} = 157874467 Lb_{7} = 14928602309 Lb_{8} = 1741809491235 Lb_{9} = 244735956424795 Lb_{10} = 40624759074089022

Table 3: Laguerre-type Bell numbers _{L}b_{n}, for n = 1, 2, …, 10.

2Lb_{1} = b_{1} + 3b_{2} + b_{3} = 12 2Lb_{2} = 4b_{2} + 32b_{3} + 38b_{4} + 12b_{5} + b_{6} = 1565 2Lb_{3} = 36b_{3} + 540b_{4} + 1242b_{5} + 882b_{6} + 243b_{7} + 27b_{8} + b_{9} = 597948 2Lb_{4} = 576b_{4} + 13824b_{5} + 50688b_{6} + 59904b_{7} + 30024b_{8} + 7200b_{9}+

+856b_{10} + 48b_{11} + b_{12} = 476170277 2Lb_{5} = 665045751420 2Lb_{6} = 1466218536786553 2Lb_{7} = 4751410403456508380 2Lb_{8} = 21492464638761800105545 2Lb_{9} = 130485451947856798889765548 2Lb_{10} = 1031029017676447641584236719317

Table 4: Laguerre-type Bell numbers _{2L}b_{n}, for n = 1, 2, …, 10

3Lb_{1} = b_{1} + 7b_{2} + 6b_{3} + b_{4} = 60 3Lb_{2} = 8b_{2} + 208b_{3} + 652b_{4} + 576b_{5} + 188b_{6} + 24b_{7} + b_{8} = 104140 3Lb_{3} = 216b_{3} + 13608b_{4} + 94284b_{5} + 186876b_{6} + 149580b_{7} + 56808b_{8}+

+11025b_{9} + 1017b_{10} + 54b_{11} + b_{12} = 811796953 3Lb_{4} = 17520336532435 3Lb_{5} = 821820979710053847 3Lb_{6} = 72322447015782400673157 3Lb_{7} = 10811794473114190401496880596 3Lb_{8} = 2555589594863343375850080281766604 3Lb_{9} = 904589617839233661347244697559337192973 3Lb_{10} = 459477819180446167103818476324632065253323967

Table 5: Laguerre-type Bell numbers _{3L}b_{n}, for n = 1, 2, …, 10

and so on, for the Laguerre-type Bell numbers of higher order.

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2010 Mathematics Subject Classification: Primary 05A10; Secondary 26A06, 11P81. Keywords: higher order Bell polynomial and number, combinatorial analysis, partition.

(Concerned with sequences \underline{A000326}, \underline{A005945}, and \underline{A144150}.)

Received March 13 2017; revised versions received August 24 2017; September 20 2017. Published in Journal of Integer Sequences, October 29 2017.

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