A378318 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*r+k,n)/(3*r+k) for k > 0.

1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 16, 30, 0, 1, 8, 30, 84, 170, 0, 1, 10, 48, 170, 496, 1050, 0, 1, 12, 70, 296, 1050, 3140, 6846, 0, 1, 14, 96, 470, 1920, 6846, 20832, 46374, 0, 1, 16, 126, 700, 3210, 12936, 46374, 142932, 323154, 0, 1, 18, 160, 994, 5040, 22402, 89712, 323154, 1005856, 2301618, 0

Offset

0,5

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 + 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 A366266.

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

Example

Square array begins:
  1,    1,     1,     1,     1,      1,      1, ...
  0,    2,     4,     6,     8,     10,     12, ...
  0,    6,    16,    30,    48,     70,     96, ...
  0,   30,    84,   170,   296,    470,    700, ...
  0,  170,   496,  1050,  1920,   3210,   5040, ...
  0, 1050,  3140,  6846, 12936,  22402,  36492, ...
  0, 6846, 20832, 46374, 89712, 159390, 266800, ...

Prog

(PARI) T(n, k, t=0, 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, A366266.

Main diagonal gives A378378.

Cf. A266213, A378317.

Cf. A378323.

Sequence in context: A229223 A128749 A106579 * A287318 A329020 A351640

Adjacent sequences: A378314 A378316 A378317 * A378319 A378320 A378321

Keyword

nonn,tabl

Author

Seiichi Manyama, Nov 23 2024

Status

approved