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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.

%I A088925 #9 Nov 18 2013 10:03:46
%S A088925 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 A088925 1,1,28,231,644,644,231,28,1,1,36,406,1652,2617,1652,406,36,1,1,45,
%U A088925 666,3738,8685,8685,3738,666,45,1,1,55,1035,7680,24735,36345,24735,7680
%N 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.
%C A088925 The g.f. for A001764 satisfies: g(x) = 1 + x*g(x)^3.
%F A088925 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 A088925 Rows begin:
%e A088925 {1, 1, 1, 1, 1, 1, 1, 1,..}
%e A088925 {1, 3, 6,10,15,21,28,..}
%e A088925 {1, 6,21,55,120,231,..}
%e A088925 {1,10,55,212,644,..}
%e A088925 {1,15,120,644,..}
%e A088925 {1,21,231,..}
%t A088925 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 A088925 Cf. A088926 (diagonal), A088927 (antidiagonal sums), A086617, A001764.
%K A088925 nonn,tabl
%O A088925 0,5
%A A088925 _Paul D. Hanna_, Oct 23 2003