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A279636 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the exponential transform of the k-th powers.

%I A279636 #16 Feb 19 2017 05:28:07
%S A279636 1,1,1,1,1,2,1,1,3,5,1,1,5,10,15,1,1,9,22,41,52,1,1,17,52,125,196,203,
%T A279636 1,1,33,130,413,836,1057,877,1,1,65,340,1445,3916,6277,6322,4140,1,1,
%U A279636 129,922,5261,19676,41077,52396,41393,21147,1,1,257,2572,19685,104116,288517,481384,479593,293608,115975
%N A279636 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the exponential transform of the k-th powers.
%H A279636 Alois P. Heinz, <a href="/A279636/b279636.txt">Antidiagonals n = 0..140, flattened</a>
%H A279636 Wikipedia, <a href="https://en.wikipedia.org/wiki/Kronecker_delta">Kronecker delta</a>
%F A279636 E.g.f. of column k: exp(exp(x)*(Sum_{j=0..k} Stirling2(n,j)*x^j) - delta_{0,k}).
%e A279636 Square array A(n,k) begins:
%e A279636 :   1,    1,    1,     1,      1,       1,        1, ...
%e A279636 :   1,    1,    1,     1,      1,       1,        1, ...
%e A279636 :   2,    3,    5,     9,     17,      33,       65, ...
%e A279636 :   5,   10,   22,    52,    130,     340,      922, ...
%e A279636 :  15,   41,  125,   413,   1445,    5261,    19685, ...
%e A279636 :  52,  196,  836,  3916,  19676,  104116,   572036, ...
%e A279636 : 203, 1057, 6277, 41077, 288517, 2133397, 16379797, ...
%p A279636 egf:= k-> exp(exp(x)*add(Stirling2(k, j)*x^j, j=0..k)-`if`(k=0, 1, 0)):
%p A279636 A:= (n, k)-> n!*coeff(series(egf(k), x, n+1), x, n):
%p A279636 seq(seq(A(n, d-n), n=0..d), d=0..12);
%p A279636 # second Maple program:
%p A279636 A:= proc(n, k) option remember; `if`(n=0, 1,
%p A279636       add(binomial(n-1, j-1)*j^k*A(n-j, k), j=1..n))
%p A279636     end:
%p A279636 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A279636 A[n_, k_] := A[n, k] = If[n==0, 1, Sum[Binomial[n-1, j-1]*j^k*A[n-j, k], {j, 1, n}]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Feb 19 2017, translated from Maple *)
%Y A279636 Columns k=0-10 give: A000110, A000248, A033462, A279358, A279637, A279638, A279639, A279640, A279641, A279642, A279643.
%Y A279636 Rows n=0+1,2 give: A000012, A000051.
%Y A279636 Main diagonal gives A279644.
%Y A279636 Cf. A145460.
%K A279636 nonn,tabl
%O A279636 0,6
%A A279636 _Alois P. Heinz_, Dec 16 2016