A378320 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(2*n-2*r+k,r) * binomial(r,n-r)/(2*n-2*r+k) for k > 0.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 2, 0, 1, 4, 6, 6, 3, 0, 1, 5, 10, 13, 11, 6, 0, 1, 6, 15, 24, 27, 22, 11, 0, 1, 7, 21, 40, 55, 57, 44, 22, 0, 1, 8, 28, 62, 100, 124, 121, 90, 44, 0, 1, 9, 36, 91, 168, 241, 278, 258, 187, 90, 0, 1, 10, 45, 128, 266, 432, 570, 620, 555, 392, 187, 0

Offset

0,8

Links

Table of n, a(n) for n=0..77.

Formula

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

G.f. of column k: B(x)^k where B(x) is the g.f. of A007477.

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

Example

Square array begins:
  1,  1,  1,   1,   1,   1,    1, ...
  0,  1,  2,   3,   4,   5,    6, ...
  0,  1,  3,   6,  10,  15,   21, ...
  0,  2,  6,  13,  24,  40,   62, ...
  0,  3, 11,  27,  55, 100,  168, ...
  0,  6, 22,  57, 124, 241,  432, ...
  0, 11, 44, 121, 278, 570, 1077, ...

Prog

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

Crossrefs

Columns k=0..1 give A000007, A007477.

Cf. A071919, A378321.

Sequence in context: A094266 A286335 A291652 * A071569 A378321 A261835

Adjacent sequences: A378317 A378318 A378319 * A378321 A378322 A378323

Keyword

nonn,tabl

Author

Seiichi Manyama, Nov 23 2024

Status

approved