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%I #3 Mar 30 2012 18:37:09
%S 1,5,29,186,1281,9294,70109,544833,4333381,35108351,288738813,
%T 2404256945,20228988678,171716799066,1468804301441,12647321103329,
%U 109538312419238,953622158606749,8340394595266367,73247287493299642
%N The first lower diagonal in square array A137570; equals the convolution of the main diagonal A137571 with the Catalan numbers (A000108) and with the square of A002293.
%F G.f. A(x) = C(x)*F(x)^2/(1 - x*C(x)*F(x)^2 - x*F(x)^3), where C(x) = 1 + xC(x)^2 is g.f. of Catalan numbers (A000108) and F(x) = 1 + xF(x)^4 is g.f. of A002293.
%e G.f.: A(x) = 1 + 5*x + 29*x^2 + 186*x^3 + 1281*x^4 + 9294*x^5 +...;
%e A(x) = C(x)*F(x)^2/(1 - x*C(x)*F(x)^2 - x*F(x)^3), where
%e C(x) = 1 + xC(x)^2 is g.f. of Catalan numbers (A000108):
%e [1, 1, 2, 5, 14, 42, 132, 429, 1430, ..., C(2n,n)/(n+1), ...] and
%e F(x) = 1 + xF(x)^4 is g.f. of A002293:
%e [1, 1, 4, 22, 140, 969, 7084, 53820, ..., C(4n,n)/(3n+1), ...].
%o (PARI) {a(n)=local(m=n+1,C,F,A); C=Ser(vector(m,r,binomial(2*r-2,r-1)/r)); F=Ser(vector(m,r,binomial(4*r-4,r-1)/(3*r-2))); A=C*F^2/(1-x*C*F^2-x*F^3);polcoeff(A+O(x^m),n,x)}
%Y Cf. A137570, A137571, A137572; A000108, A002293.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 27 2008
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