A378289 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+k,r) * binomial(r,n-r)/(n+r+k) for k > 0.

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 10, 0, 1, 4, 12, 26, 38, 0, 1, 5, 18, 49, 105, 154, 0, 1, 6, 25, 80, 210, 444, 654, 0, 1, 7, 33, 120, 363, 927, 1944, 2871, 0, 1, 8, 42, 170, 575, 1672, 4191, 8734, 12925, 0, 1, 9, 52, 231, 858, 2761, 7810, 19305, 40040, 59345, 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)^(2/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 A001002.

B(x)^k = B(x)^(k-1) + x * B(x)^(k+1) + x^2 * B(x)^(k+2). So T(n,k) = T(n,k-1) + T(n-1,k+1) + 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,   3,    7,   12,   18,    25,    33, ...
  0,  10,   26,   49,   80,   120,   170, ...
  0,  38,  105,  210,  363,   575,   858, ...
  0, 154,  444,  927, 1672,  2761,  4290, ...
  0, 654, 1944, 4191, 7810, 13325, 21385, ...

Prog

(PARI) T(n, k, t=2, 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..3 give A000007, A001002, A052706(n+2), A052703(n+3).

Cf. A009766, A026300, A378290, A378291, A378292.

Sequence in context: A106450 A255961 A297328 * A362079 A378292 A055137

Adjacent sequences: A378281 A378287 A378288 * A378290 A378291 A378292

Keyword

nonn,tabl,new

Author

Seiichi Manyama, Nov 21 2024

Status

approved