A362079 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = [x^n] 1/(1 - x*(1+x)^n)^k.

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 10, 0, 1, 4, 12, 28, 45, 0, 1, 5, 18, 55, 145, 251, 0, 1, 6, 25, 92, 315, 896, 1624, 0, 1, 7, 33, 140, 571, 2106, 6328, 11908, 0, 1, 8, 42, 200, 930, 4076, 15946, 50212, 97545, 0, 1, 9, 52, 273, 1410, 7026, 32718, 134730, 441489, 880660, 0

Offset

0,8

Links

Table of n, a(n) for n=0..65.

Formula

T(n,k) = Sum_{j=0..n} (-1)^j * binomial(-k,j) * binomial(n*j,n-j) = Sum_{j=0..n} binomial(j+k-1,j) * binomial(n*j,n-j).

Example

Square array begins:
  1,   1,   1,    1,    1,    1, ...
  0,   1,   2,    3,    4,    5, ...
  0,   3,   7,   12,   18,   25, ...
  0,  10,  28,   55,   92,  140, ...
  0,  45, 145,  315,  571,  930, ...
  0, 251, 896, 2106, 4076, 7026, ...

Prog

(PARI) T(n, k) = sum(j=0, n, binomial(j+k-1, j)*binomial(n*j, n-j));

Crossrefs

Columns k=0..3 give A000007, A099237, A362084, A362085.

Main diagonal gives A362080.

Cf. A362078.

Sequence in context: A255961 A297328 A378289 * A378292 A055137 A143325

Adjacent sequences: A362076 A362077 A362078 * A362080 A362081 A362082

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 08 2023

Status

approved