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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)).

%I A189233 #40 Dec 23 2021 06:02:15
%S A189233 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 A189233 454,309,116,30,6,1,0,877,2430,1866,756,205,42,7,1,0,4140,14214,12351,
%U A189233 5428,1555,330,56,8,1,0,21147,89918,88563,42356,12880,2850,497,72,9,1
%N 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)).
%C A189233 A(n,k) is the n-th moment of a Poisson distribution with mean = k. - _Geoffrey Critzer_, Dec 23 2018
%H A189233 Vincenzo Librandi, <a href="/A189233/b189233.txt">Table of n, a(n) for n = 0..5150</a>
%H A189233 E. T. Bell, <a href="http://www.jstor.org/stable/2300300">Exponential numbers</a>, Amer. Math. Monthly, 41 (1934), 411-419.
%H A189233 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SetPartitions">Set partitions and Bell numbers</a>
%F A189233 E.g.f. of column k: exp(k*(e^x-1)).
%F A189233 A(n,1) = A000110(n), A(n, -1) = A000587(n).
%F A189233 A(n,k) = BellPolynomial(n, k). - _Geoffrey Critzer_, Dec 23 2018
%F A189233 A(n,k) = Sum_{i=0..n} Stirling2(n,i)*k^i. - _Vladimir Kruchinin_, Apr 12 2019
%e A189233 Square array begins:
%e A189233        A000007 A000110 A001861 A027710 A078944 A144180 A144223 A144263
%e A189233 A000012   1,    1,    1,    1,    1,     1,     1,     1, ...
%e A189233 A001477   0,    1,    2,    3,    4,     5,     6,     7, ...
%e A189233 A002378   0,    2,    6,   12,   20,    30,    42,    56, ...
%e A189233 A033445   0,    5,   22,   57,  116,   205,   330,   497, ...
%e A189233           0,   15,   94,  309,  756,  1555,  2850,  4809, ...
%e A189233           0,   52,  454, 1866, 5428, 12880, 26682, 50134, ...
%p A189233 # Cf. also the Maple prog. of _Alois P. Heinz_ in A144223 and A144180.
%p A189233 expnums := proc(k,n) option remember; local j;
%p A189233 `if`(n = 0, 1, (1+add(binomial(n-1,j-1)*expnums(k,n-j), j = 1..n-1))*k) end:
%p A189233 A189233_array := (k, n) -> expnums(k,n):
%p A189233 seq(print(seq(A189233_array(k,n), k = 0..7)), n = 0..5);
%p A189233 A189233_egf := k -> exp(k*(exp(x)-1));
%p A189233 T := (n,k) -> n!*coeff(series(A189233_egf(k), x, n+1), x, n):
%p A189233 seq(lprint(seq(T(n,k), k = 0..7)), n = 0..5):
%p A189233 # alternative Maple program:
%p A189233 A:= proc(n, k) option remember; `if`(n=0, 1,
%p A189233       (1+add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
%p A189233     end:
%p A189233 seq(seq(A(d-k, k), k=0..d), d=0..12);  # _Alois P. Heinz_, Sep 25 2017
%t A189233 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 A189233 Table[Table[BellB[n, k], {k, 0, 5}], {n, 0, 5}] // Grid  (* _Geoffrey Critzer_, Dec 23 2018 *)
%o A189233 (Maxima)
%o A189233 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 A189233 Columns: A000007, A000110, A001861, A027710, A078944, A144180, A144223, A144263.
%Y A189233 Rows: A000012, A001477, A002378, A033445.
%Y A189233 Main diagonal gives A242817.
%Y A189233 Cf. A000587, A144150.
%K A189233 nonn,tabl
%O A189233 0,8
%A A189233 _Peter Luschny_, Apr 18 2011