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A383049 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) is the n-th term of the inverse Stirling transform of j-> (j+1)^k.

%I A383049 #17 Apr 15 2025 08:25:42
%S A383049 1,1,1,1,2,0,1,4,1,0,1,8,5,-1,0,1,16,19,-3,2,0,1,32,65,-1,4,-6,0,1,64,
%T A383049 211,45,-10,-8,24,0,1,128,665,359,-116,48,20,-120,0,1,256,2059,2037,
%U A383049 -538,340,-234,-52,720,0,1,512,6305,10079,-1316,984,-1240,1302,72,-5040,0
%N A383049 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) is the n-th term of the inverse Stirling transform of j-> (j+1)^k.
%H A383049 Christian G. Bower, <a href="https://oeis.org/transforms_pari.txt">PARI programs for transforms</a>, 2007.
%H A383049 N. J. A. Sloane, <a href="/transforms.txt">Maple programs for transforms</a>, 2001-2020.
%F A383049 A(n,k) = Sum_{j=0..n} (j+1)^k * Stirling1(n,j).
%F A383049 E.g.f. of column k: Sum_{j>=0} (j+1)^k * log(1+x)^j / j!.
%F A383049 E.g.f. of column k: (1+x) * Sum_{j=0..k} Stirling2(k+1,j+1) * log(1+x)^j.
%e A383049 Square array begins:
%e A383049   1,  1,  1,    1,     1,     1,     1, ...
%e A383049   1,  2,  4,    8,    16,    32,    64, ...
%e A383049   0,  1,  5,   19,    65,   211,   665, ...
%e A383049   0, -1, -3,   -1,    45,   359,  2037, ...
%e A383049   0,  2,  4,  -10,  -116,  -538, -1316, ...
%e A383049   0, -6, -8,   48,   340,   984, -1148, ...
%e A383049   0, 24, 20, -234, -1240, -1866, 16400, ...
%o A383049 (PARI) a(n, k) = sum(j=0, n, (j+1)^k*stirling(n, j, 1));
%Y A383049 Columns k=0..6 give A019590(n+1), A302190 (for n > 0), A222627, A222636, A222748, A223023, A383050.
%Y A383049 Main diagonal gives A383051.
%Y A383049 Cf. A362924, A362925.
%K A383049 sign,tabl
%O A383049 0,5
%A A383049 _Seiichi Manyama_, Apr 14 2025