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A189233
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Square array A(n,k), n >= 0, k >= 0, read by antidiagonals upwards, where the e.g.f. of column k is exp(k*(e^x-1)).
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16
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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, 0, 21147, 89918, 88563, 42356, 12880, 2850, 497, 72, 9, 1
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OFFSET
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0,8
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COMMENTS
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A(n,k) is the n-th moment of a Poisson distribution with mean = k. - Geoffrey Critzer, Dec 23 2018
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LINKS
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FORMULA
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E.g.f. of column k: exp(k*(e^x-1)).
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EXAMPLE
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Square array begins:
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, ...
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MAPLE
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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):
# alternative Maple program:
A:= proc(n, k) option remember; `if`(n=0, 1,
(1+add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
end:
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MATHEMATICA
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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-François Alcover, Jun 26 2013 *)
Table[Table[BellB[n, k], {k, 0, 5}], {n, 0, 5}] // Grid (* Geoffrey Critzer, Dec 23 2018 *)
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PROG
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(Maxima)
A(n, k):=if k=0 and n=0 then 1 else if k=0 then 0 else sum(stirling2(n, i)*k^i, i, 0, n); /* Vladimir Kruchinin, Apr 12 2019 */
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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