A378238 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(3*n+r+k,n)/(3*n+r+k) for k > 0.

1, 1, 0, 1, 2, 0, 1, 4, 14, 0, 1, 6, 32, 134, 0, 1, 8, 54, 324, 1482, 0, 1, 10, 80, 578, 3696, 17818, 0, 1, 12, 110, 904, 6810, 45316, 226214, 0, 1, 14, 144, 1310, 11008, 85278, 583152, 2984206, 0, 1, 16, 182, 1804, 16490, 140936, 1113854, 7769348, 40503890, 0

Offset: 0

Author

Seiichi Manyama, Nov 20 2024

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1,      1,      1,       1,       1,       1,       1, ...
  0,      2,      4,       6,       8,      10,      12, ...
  0,     14,     32,      54,      80,     110,     144, ...
  0,    134,    324,     578,     904,    1310,    1804, ...
  0,   1482,   3696,    6810,   11008,   16490,   23472, ...
  0,  17818,  45316,   85278,  140936,  216002,  314700, ...
  0, 226214, 583152, 1113854, 1870352, 2914790, 4320608, ...
		

Crossrefs

Columns k=0..3 give A000007, A144097, A371675, A365843.

T(n,n) gives 1/4 * A370102(n) for n > 0.

Cf. A033877, A071949, A378236, A378237, A378239, A378240.

Programs

PARI

T(n, k, t=3, u=1) = 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 * A_k(x)^(3/k) * (1 + A_k(x)^(1/k)) )^k for k > 0.

G.f. of column k: B(x)^k where B(x) is the g.f. of A144097.

B(x)^k = B(x)^(k-1) + x * B(x)^(k+2) + x * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k+2) + T(n-1,k+3) for n > 0.