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A172391 G.f. satisfies: A(x) = G(x/A(x))^2 and G(x)^2 = A(x*G(x)^2) where G(x) = Sum_{n>=0} C(2*n,n)*C(2*n+2,n+1)/(n+2)*x^n is the g.f. of A172392.

%I A172391 #7 Feb 08 2025 01:23:08
%S A172391 1,8,12,0,28,0,264,0,3720,0,63840,0,1232432,0,25731216,0,568130552,0,
%T A172391 13081215840,0,311178567648,0,7597974517056,0,189518147463232,0,
%U A172391 4811962763222784,0,124028853694440640,0,3238304402221646880,0
%N A172391 G.f. satisfies: A(x) = G(x/A(x))^2 and G(x)^2 = A(x*G(x)^2) where G(x) = Sum_{n>=0} C(2*n,n)*C(2*n+2,n+1)/(n+2)*x^n is the g.f. of A172392.
%F A172391 G.f.: A(x) = x/Series_Reversion(x*G(x)^2) where G(x) is the g.f. of A172392(n) = A000108(n+1)*A000984(n).
%F A172391 Self-convolution of A172393.
%e A172391 G.f.: A(x) = 1 + 8*x + 12*x^2 + 28*x^4 + 264*x^6 + 3720*x^8 +...
%e A172391 where A(x) = G(x/A(x))^2 where G(x) is the g.f. of A172392:
%e A172391 G(x) = 1 + 4*x + 30*x^2 + 280*x^3 + 2940*x^4 + 33264*x^5 +...+ A172392(n)*x^n +...
%e A172391 G(x) = 1 + 2*2*x + 5*6*x^2 + 14*20*x^3 + 42*70*x^4 + 132*252*x^5 +...
%o A172391 (PARI) {a(n)=local(G=sum(m=0,n,binomial(2*m,m)*binomial(2*m+2,m+1)/(m+2)*x^m)+x*O(x^n));polcoeff(x/serreverse(x*G^2),n)}
%Y A172391 Cf. A000108, A000984, A172392, A172393, variants: A172390, A168357, A168451, A168452.
%K A172391 nonn
%O A172391 0,2
%A A172391 _Paul D. Hanna_, Feb 05 2010