A088925 Square table, read by antidiagonals, of coefficients T(n,k) of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xy*f(x,y)^3.

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 21, 10, 1, 1, 15, 55, 55, 15, 1, 1, 21, 120, 212, 120, 21, 1, 1, 28, 231, 644, 644, 231, 28, 1, 1, 36, 406, 1652, 2617, 1652, 406, 36, 1, 1, 45, 666, 3738, 8685, 8685, 3738, 666, 45, 1, 1, 55, 1035, 7680, 24735, 36345, 24735, 7680

Offset

0,5

Comments

The g.f. for A001764 satisfies: g(x) = 1 + x*g(x)^3.

Links

Table of n, a(n) for n=0..62.

Formula

T(n, k) = sum(i=0, k, C(n+k, 2i)*C(n+k-2i, k-i)*A001764(i) ), where A001764(i)=(3i)!/(i!(2i+1)!). - from Michael Somos

Example

Rows begin:
{1, 1, 1, 1, 1, 1, 1, 1,..}
{1, 3, 6,10,15,21,28,..}
{1, 6,21,55,120,231,..}
{1,10,55,212,644,..}
{1,15,120,644,..}
{1,21,231,..}

Mathematica

t[n_, k_] := Sum[ Binomial[n+k, 2*i]*Binomial[n+k-2*i, k-i]*(3*i)!/(i!*(2*i+1)!), {i, 0, k}]; Table[t[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 18 2013, after Michael Somos *)

Crossrefs

Cf. A088926 (diagonal), A088927 (antidiagonal sums), A086617, A001764.

Sequence in context: A001263 A162747 A107105 * A100862 A098568 A180959

Adjacent sequences: A088922 A088923 A088924 * A088926 A088927 A088928

Keyword

nonn,tabl

Author

Paul D. Hanna, Oct 23 2003

Status

approved