A189233 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)).

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

Offset

0,8

Comments

A(n,k) is the n-th moment of a Poisson distribution with mean = k. - Geoffrey Critzer, Dec 23 2018

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..5150

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

Peter Luschny, Set partitions and Bell numbers

Formula

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

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

A(n,k) = BellPolynomial(n, k). - Geoffrey Critzer, Dec 23 2018

A(n,k) = Sum_{i=0..n} Stirling2(n,i)*k^i. - Vladimir Kruchinin, Apr 12 2019

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):
# 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:
seq(seq(A(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Sep 25 2017

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-François Alcover, Jun 26 2013 *)
Table[Table[BellB[n, k], {k, 0, 5}], {n, 0, 5}] // Grid  (* Geoffrey Critzer, Dec 23 2018 *)

Prog

(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 */

Crossrefs

Columns: A000007, A000110, A001861, A027710, A078944, A144180, A144223, A144263.

Rows: A000012, A001477, A002378, A033445.

Main diagonal gives A242817.

Cf. A000587, A144150.

Sequence in context: A067347 A120568 A321960 * A242153 A065066 A266291

Adjacent sequences: A189230 A189231 A189232 * A189234 A189235 A189236

Keyword

nonn,tabl

Author

Peter Luschny, Apr 18 2011

Status

approved