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%I #7 Oct 10 2020 03:04:07
%S 1,3,21,212,2617,36345,544080,8577378,140456625,2368062095,
%T 40859183247,718386164556,12829418522056,232153200359592,
%U 4248457201595622,78508329463480160,1463164022514939392,27474112707608092672
%N Main diagonal of table A088925, which lists 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 a(n) = sum(i=0, n, C(2n, 2i)*C(2n-2i, n-i)*A001764(i) ), where A001764(i)=(3i)!/[i!(2i+1)! ] (from Michael Somos).
%F a(n) ~ (4 + 3*sqrt(3))^(2*n + 2) / (Pi * 3^(7/4) * n^2 * 2^(2*n + 4)). - _Vaclav Kotesovec_, Oct 10 2020
%t Table[Sum[Binomial[2*n, 2*i] * Binomial[2*n - 2*i, n - i]*(3*i)!/(i!*(2*i + 1)!), {i, 0, n}], {n, 0, 25}] (* _Vaclav Kotesovec_, Oct 10 2020 *)
%Y Cf. A088925 (table), A088927 (antidiagonal sums), A001764.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Oct 23 2003
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