A368506 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * binomial(j+k-1,j).

1, 1, 0, 1, 2, 0, 1, 4, 3, 0, 1, 6, 11, 4, 0, 1, 8, 24, 26, 5, 0, 1, 10, 42, 82, 57, 6, 0, 1, 12, 65, 188, 261, 120, 7, 0, 1, 14, 93, 360, 787, 804, 247, 8, 0, 1, 16, 126, 614, 1870, 3204, 2440, 502, 9, 0, 1, 18, 164, 966, 3810, 9476, 12900, 7356, 1013, 10, 0

Offset: 0

Author

Seiichi Manyama, Dec 27 2023

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1, 1,   1,    1,     1,     1,      1, ...
  0, 2,   4,    6,     8,    10,     12, ...
  0, 3,  11,   24,    42,    65,     93, ...
  0, 4,  26,   82,   188,   360,    614, ...
  0, 5,  57,  261,   787,  1870,   3810, ...
  0, 6, 120,  804,  3204,  9476,  23112, ...
  0, 7, 247, 2440, 12900, 47590, 139134, ...
		

Crossrefs

Columns k=0..3 give A000007, A000027(n+1), A125128(n+1), A052150.

Main diagonal gives A293574.

Cf. A008949, A368487.

Programs

PARI

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

Formula

G.f. of column k: 1/((1-k*x) * (1-x)^k).