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%I #30 May 12 2021 06:19:09
%S 1,1,0,1,1,1,1,2,2,1,1,3,5,6,8,1,4,10,15,24,26,1,5,17,34,54,120,194,1,
%T 6,26,69,104,240,720,1142,1,7,37,126,204,200,1350,5040,9736,1,8,50,
%U 211,408,-330,-400,9450,40320,81384,1,9,65,330,794,-1704,-12510,-2800,78120,362880,823392
%N Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. log(1 + Sum_{j>=1} binomial(j+k-1,k) * x^j/j).
%C Column k > 2 is asymptotic to -2*(n-1)! * cos(n*arctan(sin(Pi/k)/(cos(Pi/k) - (k-1)^(1/k)))) / (1 + 1/(k-1)^(2/k) - 2*cos(Pi/k)/(k-1)^(1/k))^(n/2). - _Vaclav Kotesovec_, May 12 2021
%H Seiichi Manyama, <a href="/A308497/b308497.txt">Antidiagonals n = 1..140, flattened</a>
%F A(n,k) = (1/k!) * ((n+k-1)! - Sum_{j=1..n-1} binomial(n-1,j) * (j+k-1)! * A(n-j,k)).
%F E.g.f.: log(1 + (1/(1-x)^k - 1)/k). - _Vaclav Kotesovec_, May 12 2021
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, ...
%e 1, 2, 5, 10, 17, 26, ...
%e 1, 6, 15, 34, 69, 126, ...
%e 8, 24, 54, 104, 204, 408, ...
%e 26, 120, 240, 200, -330, -1704, ...
%e 194, 720, 1350, -400, -12510, -51696, ...
%t T[n_, k_] := T[n, k] = ((n+k-1)! - Sum[Binomial[n-1,j] * (j+k-1)! * T[n-j,k], {j,1,n-1}])/k!; Table[T[k, n - k], {n, 1, 11}, {k, 1, n}] // Flatten (* _Amiram Eldar_, May 12 2021 *)
%Y Columns k=0..5 give A089064, A000142(n-1), (-1)^(n+1) * A009383(n), A308499, A344217, A344218.
%K sign,tabl
%O 1,8
%A _Seiichi Manyama_, Jun 01 2019
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