A378240 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,r) * binomial(3*n+3*r+k,n)/(3*n+3*r+k) for k > 0.

1, 1, 0, 1, 2, 0, 1, 4, 18, 0, 1, 6, 40, 234, 0, 1, 8, 66, 540, 3570, 0, 1, 10, 96, 926, 8400, 59586, 0, 1, 12, 130, 1400, 14706, 141876, 1053570, 0, 1, 14, 168, 1970, 22720, 251622, 2528760, 19392490, 0, 1, 16, 210, 2644, 32690, 394152, 4524786, 46815116, 367677090, 0

Offset

0,5

Links

Table of n, a(n) for n=0..54.

Formula

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(3/k) * (1 + 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 A364167.

B(x)^k = B(x)^(k-1) + x * B(x)^(k+2) + x * B(x)^(k+5). So T(n,k) = T(n,k-1) + T(n-1,k+2) + T(n-1,k+5) for n > 0.

Example

Square array begins:
  1,       1,       1,       1,       1,        1,        1, ...
  0,       2,       4,       6,       8,       10,       12, ...
  0,      18,      40,      66,      96,      130,      168, ...
  0,     234,     540,     926,    1400,     1970,     2644, ...
  0,    3570,    8400,   14706,   22720,    32690,    44880, ...
  0,   59586,  141876,  251622,  394152,   575402,   801948, ...
  0, 1053570, 2528760, 4524786, 7156128, 10553970, 14867704, ...

Prog

(PARI) T(n, k, t=3, u=3) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

Crossrefs

Columns k=0..1 give A000007, A364167.

Cf. A033877, A071949, A378236, A378237, A378238, A378239.

Sequence in context: A293961 A378239 A378238 * A364228 A112899 A212808

Adjacent sequences: A378237 A378238 A378239 * A378241 A378242 A378243

Keyword

nonn,tabl

Author

Seiichi Manyama, Nov 20 2024

Status

approved