A386558 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = binomial((k+1)*n+k-1,n)/(n+1).

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 5, 0, 1, 4, 15, 30, 14, 0, 1, 5, 26, 91, 143, 42, 0, 1, 6, 40, 204, 612, 728, 132, 0, 1, 7, 57, 385, 1771, 4389, 3876, 429, 0, 1, 8, 77, 650, 4095, 16380, 32890, 21318, 1430, 0, 1, 9, 100, 1015, 8184, 46376, 158224, 254475, 120175, 4862, 0

Offset: 0

Author

Seiichi Manyama, Jul 26 2025

Keywords

• nonn
• tabl
• easy

Examples

			Square array begins:
  1,   1,    1,     1,      1,      1,       1, ...
  0,   1,    2,     3,      4,      5,       6, ...
  0,   2,    7,    15,     26,     40,      57, ...
  0,   5,   30,    91,    204,    385,     650, ...
  0,  14,  143,   612,   1771,   4095,    8184, ...
  0,  42,  728,  4389,  16380,  46376,  109668, ...
  0, 132, 3876, 32890, 158224, 548340, 1533939, ...
		

Links

• Wikipedia, Fuss-Catalan number

Crossrefs

Columns k=0..10 give A000007, A000108, A006013, A006632, A118971, A130564(n+1), A130565(n+1), A234466, A234513, A234573, A235340.

Main diagonal gives A177784(n+1).

Cf. A162382.

Programs

PARI

a(n, k) = binomial((k+1)*n+k-1, n)/(n+1);

Formula

For k > 0, A(n,k) = r * binomial(n*p+r,n)/(n*p+r), the Fuss-Catalan number with p=k+1 and r=k.

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