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A291556 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) = (n!)^k * Sum_{i=1..n} 1/i^k.

%I A291556 #26 May 11 2019 10:10:57
%S A291556 0,0,1,0,1,2,0,1,3,3,0,1,5,11,4,0,1,9,49,50,5,0,1,17,251,820,274,6,0,
%T A291556 1,33,1393,16280,21076,1764,7,0,1,65,8051,357904,2048824,773136,13068,
%U A291556 8,0,1,129,47449,8252000,224021776,444273984,38402064,109584,9
%N A291556 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) = (n!)^k * Sum_{i=1..n} 1/i^k.
%H A291556 Seiichi Manyama, <a href="/A291556/b291556.txt">Antidiagonals n = 0..59, flattened</a>
%F A291556 A(0, k) = 0, A(1, k) = 1, A(n+1, k) = (n^k+(n+1)^k)*A(n, k) - n^(2*k)*A(n-1, k).
%e A291556 Square array begins:
%e A291556    0,  0,   0,     0,      0, ...
%e A291556    1,  1,   1,     1,      1, ...
%e A291556    2,  3,   5,     9,     17, ...
%e A291556    3, 11,  49,   251,   1393, ...
%e A291556    4, 50, 820, 16280, 357904, ...
%p A291556 A:= (n, k)-> n!^k * add(1/i^k, i=1..n):
%p A291556 seq(seq(A(n, d-n), n=0..d), d=0..10);  # _Alois P. Heinz_, Aug 26 2017
%t A291556 A[0, _] = 0; A[1, _] = 1; A[n_, k_] := A[n, k] = ((n-1)^k + n^k) A[n-1, k] - (n-1)^(2k) A[n-2, k];
%t A291556 Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, May 11 2019 *)
%Y A291556 Columns k=0-10 give: A001477, A000254, A001819, A066989, A203229, A099827, A291456, A291505, A291506, A291507, A291508.
%Y A291556 Rows n=0-3 give: A000004, A000012, A000051, A074528.
%Y A291556 Main diagonal gives A060943.
%K A291556 nonn,tabl
%O A291556 0,6
%A A291556 _Seiichi Manyama_, Aug 26 2017