%I #9 Nov 18 2013 10:03:46

%S 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,

%T 1,1,28,231,644,644,231,28,1,1,36,406,1652,2617,1652,406,36,1,1,45,

%U 666,3738,8685,8685,3738,666,45,1,1,55,1035,7680,24735,36345,24735,7680

%N 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.

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

%F 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_

%e Rows begin:

%e {1, 1, 1, 1, 1, 1, 1, 1,..}

%e {1, 3, 6,10,15,21,28,..}

%e {1, 6,21,55,120,231,..}

%e {1,10,55,212,644,..}

%e {1,15,120,644,..}

%e {1,21,231,..}

%t 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_ *)

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

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Oct 23 2003