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

1,8

Comments

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

Links

Seiichi Manyama, Antidiagonals n = 1..140, flattened

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

Example

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, ...

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 *)

Crossrefs

Columns k=0..5 give A089064, A000142(n-1), (-1)^(n+1) * A009383(n), A308499, A344217, A344218.

Sequence in context: A274887 A008302 A131791 * A010358 A155865 A156133

Adjacent sequences: A308494 A308495 A308496 * A308498 A308499 A308500

Keyword

sign,tabl

Author

Seiichi Manyama, Jun 01 2019

Status

approved