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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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%I #40 Dec 23 2021 06:02:15

%S 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,

%T 454,309,116,30,6,1,0,877,2430,1866,756,205,42,7,1,0,4140,14214,12351,

%U 5428,1555,330,56,8,1,0,21147,89918,88563,42356,12880,2850,497,72,9,1

%N 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)).

%C A(n,k) is the n-th moment of a Poisson distribution with mean = k. - _Geoffrey Critzer_, Dec 23 2018

%H Vincenzo Librandi, <a href="/A189233/b189233.txt">Table of n, a(n) for n = 0..5150</a>

%H E. T. Bell, <a href="http://www.jstor.org/stable/2300300">Exponential numbers</a>, Amer. Math. Monthly, 41 (1934), 411-419.

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SetPartitions">Set partitions and Bell numbers</a>

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

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

%F A(n,k) = BellPolynomial(n, k). - _Geoffrey Critzer_, Dec 23 2018

%F A(n,k) = Sum_{i=0..n} Stirling2(n,i)*k^i. - _Vladimir Kruchinin_, Apr 12 2019

%e Square array begins:

%e A000007 A000110 A001861 A027710 A078944 A144180 A144223 A144263

%e A000012 1, 1, 1, 1, 1, 1, 1, 1, ...

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

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

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

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

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

%p # Cf. also the Maple prog. of _Alois P. Heinz_ in A144223 and A144180.

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

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

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

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

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

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

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

%p # alternative Maple program:

%p A:= proc(n, k) option remember; `if`(n=0, 1,

%p (1+add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)

%p end:

%p seq(seq(A(d-k, k), k=0..d), d=0..12); # _Alois P. Heinz_, Sep 25 2017

%t 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 *)

%t Table[Table[BellB[n, k], {k, 0, 5}], {n, 0, 5}] // Grid (* _Geoffrey Critzer_, Dec 23 2018 *)

%o (Maxima)

%o 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 */

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

%Y Rows: A000012, A001477, A002378, A033445.

%Y Main diagonal gives A242817.

%Y Cf. A000587, A144150.

%K nonn,tabl

%O 0,8

%A _Peter Luschny_, Apr 18 2011