A378317 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(2*r+k,n)/(2*r+k) for k > 0.

1, 1, 0, 1, 2, 0, 1, 4, 4, 0, 1, 6, 12, 12, 0, 1, 8, 24, 40, 40, 0, 1, 10, 40, 92, 144, 144, 0, 1, 12, 60, 176, 360, 544, 544, 0, 1, 14, 84, 300, 752, 1440, 2128, 2128, 0, 1, 16, 112, 472, 1400, 3200, 5872, 8544, 8544, 0, 1, 18, 144, 700, 2400, 6352, 13664, 24336, 35008, 35008, 0

Offset: 0

Author

Seiichi Manyama, Nov 23 2024

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1,   1,    1,    1,     1,     1,     1, ...
  0,   2,    4,    6,     8,    10,    12, ...
  0,   4,   12,   24,    40,    60,    84, ...
  0,  12,   40,   92,   176,   300,   472, ...
  0,  40,  144,  360,   752,  1400,  2400, ...
  0, 144,  544, 1440,  3200,  6352, 11616, ...
  0, 544, 2128, 5872, 13664, 28480, 54768, ...
		

Crossrefs

Columns k=0..1 give A000007, A025227(n+1).

Main diagonal gives A333473.

Cf. A266213, A378318.

Programs

PARI

T(n, k, t=0, u=2) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

Formula

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x + x * A_k(x)^(2/k) )^k for k > 0.

G.f. of column k: (B(x)/x)^k where B(x) is the g.f. of A025227.

B(x)^k = B(x)^(k-1) + x * B(x)^(k-1) + x * B(x)^(k+1). So T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k+1) for n > 0.