A378291 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-2*r+k,r) * binomial(r,n-r)/(3*n-2*r+k) for k > 0.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 6, 0, 1, 4, 9, 16, 20, 0, 1, 5, 14, 31, 56, 72, 0, 1, 6, 20, 52, 114, 208, 273, 0, 1, 7, 27, 80, 201, 438, 806, 1073, 0, 1, 8, 35, 116, 325, 800, 1739, 3220, 4333, 0, 1, 9, 44, 161, 495, 1341, 3260, 7077, 13168, 17869, 0

Offset

0,8

Links

Table of n, a(n) for n=0..65.

Formula

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(1/k) * (1 + x * 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 A186996.

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

Example

Square array begins:
  1,   1,   1,    1,    1,    1,    1, ...
  0,   1,   2,    3,    4,    5,    6, ...
  0,   2,   5,    9,   14,   20,   27, ...
  0,   6,  16,   31,   52,   80,  116, ...
  0,  20,  56,  114,  201,  325,  495, ...
  0,  72, 208,  438,  800, 1341, 2118, ...
  0, 273, 806, 1739, 3260, 5615, 9119, ...

Prog

(PARI) T(n, k, t=1, 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, A186996.

Cf. A009766, A026300, A378289, A378290, A378292.

Cf. A378237.

Sequence in context: A277938 A130020 A292870 * A306704 A091063 A246935

Adjacent sequences: A378288 A378289 A378290 * A378292 A378293 A378294

Keyword

nonn,tabl,new

Author

Seiichi Manyama, Nov 21 2024

Status

approved