A378321 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(3*n-3*r+k,r) * binomial(r,n-r)/(3*n-3*r+k) for k > 0.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 3, 0, 1, 4, 6, 8, 6, 0, 1, 5, 10, 16, 19, 16, 0, 1, 6, 15, 28, 42, 50, 42, 0, 1, 7, 21, 45, 79, 114, 137, 114, 0, 1, 8, 28, 68, 135, 224, 322, 380, 322, 0, 1, 9, 36, 98, 216, 401, 652, 918, 1088, 918, 0, 1, 10, 45, 136, 329, 672, 1205, 1912, 2673, 3152, 2673, 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)^(3/k) )^k for k > 0.

G.f. of column k: B(x)^k where B(x) is the g.f. of A019497.

B(x)^k = B(x)^(k-1) + x * B(x)^(k-1) + x^2 * B(x)^(k+2). So T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-2,k+2) 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,  3,   8,  16,  28,   45,   68, ...
  0,  6,  19,  42,  79,  135,  216, ...
  0, 16,  50, 114, 224,  401,  672, ...
  0, 42, 137, 322, 652, 1205, 2088, ...

Prog

(PARI) T(n, k, t=0, u=3) = 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, A019497.

Cf. A071919, A378320.

Sequence in context: A291652 A378320 A071569 * A261835 A353436 A286932

Adjacent sequences: A378318 A378319 A378320 * A378322 A378323 A378324

Keyword

nonn,tabl

Author

Seiichi Manyama, Nov 23 2024

Status

approved