A308497 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).
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, 6, 26, 69, 104, 240, 720, 1142, 1, 7, 37, 126, 204, 200, 1350, 5040, 9736, 1, 8, 50, 211, 408, -330, -400, 9450, 40320, 81384, 1, 9, 65, 330, 794, -1704, -12510, -2800, 78120, 362880, 823392
Offset: 1
Examples
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
1, 2, 5, 10, 17, 26, ...
1, 6, 15, 34, 69, 126, ...
8, 24, 54, 104, 204, 408, ...
26, 120, 240, 200, -330, -1704, ...
194, 720, 1350, -400, -12510, -51696, ...
Links
- Seiichi Manyama, Antidiagonals n = 1..140, flattened
Crossrefs
Programs
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Mathematica
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 *)
Formula
A(n,k) = (1/k!) * ((n+k-1)! - Sum_{j=1..n-1} binomial(n-1,j) * (j+k-1)! * A(n-j,k)).
E.g.f.: log(1 + (1/(1-x)^k - 1)/k). - Vaclav Kotesovec, May 12 2021
Comments