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A189233 Square array A(n,k), n >= 0, k >= 0, read by diagonals, where the e.g.f. of column k is exp(k*(e^x-1)). 9
1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 6, 3, 1, 0, 15, 22, 12, 4, 1, 0, 52, 94, 57, 20, 5, 1, 0, 203, 454, 309, 116, 30, 6, 1, 0, 877, 2430, 1866, 756, 205, 42, 7, 1, 0, 4140, 14214, 12351, 5428, 1555, 330, 56, 8, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

A(n, 1) = A000110(n), A(n, -1) = A000587(n).

REFERENCES

E. T. Bell, Exponential numbers, Amer. Math. Monthly, 41 (1934), 411-419.

LINKS

Table of n, a(n) for n=0..54.

Peter Luschny, Set partitions and Bell numbers

FORMULA

E.g.f. of column k is exp(k*(e^x-1)).

EXAMPLE

Square array begins:

       A000007 A000110 A001861 A027710 A078944 A144180 A144223 A144263

A000012    1,    1,    1,    1,    1,     1,     1,     1, ...

A001477    0,    1,    2,    3,    4,     5,     6,     7, ...

A002378    0,    2,    6,   12,   20,    30,    42,    56, ...

A033445    0,    5,   22,   57,  116,   205,   330,   497, ...

           0,   15,   94,  309,  756,  1555,  2850,  4809, ...

           0,   52,  454, 1866, 5428, 12880, 26682, 50134, ...

MAPLE

# Cf. also the Maple prog. of Alois P. Heinz in A144223 and A144180.

expnums := proc(k, n) option remember; local j;

`if`(n = 0, 1, (1+add(binomial(n-1, j-1)*expnums(k, n-j), j = 1..n-1))*k) end:

A189233_array := (k, n) -> expnums(k, n):

seq(print(seq(A189233_array(k, n), k = 0..7)), n = 0..5);

A189233_egf := k -> exp(k*(exp(x)-1));

T := (n, k) -> n!*coeff(series(A189233_egf(k), x, n+1), x, n):

seq(lprint(seq(T(n, k), k = 0..7)), n = 0..5):

MATHEMATICA

max = 9; Clear[col]; col[k_] := col[k] = CoefficientList[ Series[ Exp[k*(Exp[x]-1)], {x, 0, max}], x]*Range[0, max]!; a[0, _] = 1; a[n_?Positive, 0] = 0; a[n_, k_] := col[k][[n+1]]; Table[ a[n-k, k], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Jun 26 2013 *)

CROSSREFS

Cf. A144150.

Main diagonal gives A242817.

Sequence in context: A120059 A067347 A120568 * A242153 A065066 A266291

Adjacent sequences:  A189230 A189231 A189232 * A189234 A189235 A189236

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Apr 18 2011

STATUS

approved

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Last modified December 9 13:48 EST 2016. Contains 278971 sequences.