A378290 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+2*r+k,r) * binomial(r,n-r)/(n+2*r+k) for k > 0.

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 9, 19, 0, 1, 4, 15, 46, 104, 0, 1, 5, 22, 82, 262, 614, 0, 1, 6, 30, 128, 486, 1588, 3816, 0, 1, 7, 39, 185, 789, 3027, 10053, 24595, 0, 1, 8, 49, 254, 1185, 5052, 19543, 65686, 162896, 0, 1, 9, 60, 336, 1689, 7801, 33290, 129606, 439658, 1101922, 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)^(3/k) * (1 + x * 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 A186997.

B(x)^k = B(x)^(k-1) + x * B(x)^(k+2) + x^2 * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k+2) + 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,    4,     9,    15,    22,    30,    39, ...
  0,   19,    46,    82,   128,   185,   254, ...
  0,  104,   262,   486,   789,  1185,  1689, ...
  0,  614,  1588,  3027,  5052,  7801, 11430, ...
  0, 3816, 10053, 19543, 33290, 52490, 78552, ...

Prog

(PARI) T(n, k, t=3, u=1) = 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, A186997, A218045(n+2).

Cf. A009766, A026300, A378289, A378291, A378292.

Sequence in context: A256245 A173004 A378323 * A118343 A309148 A351761

Adjacent sequences: A378287 A378288 A378289 * A378291 A378292 A378293

Keyword

nonn,tabl,new

Author

Seiichi Manyama, Nov 21 2024

Status

approved