A378323 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(3r+k,r) * binomial(r,n-r)/(3*r+k) for k > 0.

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 9, 18, 0, 1, 4, 15, 44, 94, 0, 1, 5, 22, 79, 240, 529, 0, 1, 6, 30, 124, 450, 1390, 3135, 0, 1, 7, 39, 180, 737, 2685, 8404, 19270, 0, 1, 8, 49, 248, 1115, 4532, 16585, 52426, 121732, 0, 1, 9, 60, 329, 1599, 7066, 28624, 105147, 334964, 785496, 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 * (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 A364475.

B(x)^k = B(x)^(k-1) + x * B(x)^(k+2) + x^2 * B(x)^(k+2). So T(n,k) = T(n,k-1) + T(n-1,k+2) + 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,    4,    9,    15,    22,    30,    39, ...
  0,   18,   44,    79,   124,   180,   248, ...
  0,   94,  240,   450,   737,  1115,  1599, ...
  0,  529, 1390,  2685,  4532,  7066, 10440, ...
  0, 3135, 8404, 16585, 28624, 45655, 69021, ...

Prog

(PARI) T(n, k, t=3, u=0) = 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..2 give A000007, A364475, A371576.

Cf. A038137, A071943.

Cf. A378318.

Sequence in context: A258761 A256245 A173004 * A378290 A118343 A381602

Adjacent sequences: A378320 A378321 A378322 * A378324 A378325 A378326

Keyword

nonn,tabl

Author

Seiichi Manyama, Nov 23 2024

Status

approved