This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A172393 #2 Mar 30 2012 18:37:21
%S A172393 1,4,-2,8,-20,96,-324,1648,-6348,33200,-137848,732640,-3193296,
%T A172393 17148608,-77335400,418289696,-1934677436,10518803376,-49611450120,
%U A172393 270796872160,-1297234193744,7102371571840,-34458382484976,189117499963840
%N A172393 G.f. satisfies: A(x) = G(x/A(x)^2) and G(x) = A(x*G(x)^2) = Sum_{n>=0} C(2n,n)*C(2n+2,n+1)/(n+2)*x^n is the g.f. of A172392.
%F A172393 G.f. satisfies: A(x) = Sum_{n>=0} A000108(n+1)*A000984(n)*x^n/A(x)^(2n), where A000108 is the Catalan numbers and A000984 is the central binomial coefficients.
%F A172393 Self-convolution equals A172391.
%e A172393 G.f.: A(x) = 1 + 4*x - 2*x^2 + 8*x^3 - 20*x^4 + 96*x^5 - 324*x^6 +...
%e A172393 A(x)^2 = 1 + 8*x + 12*x^2 + 28*x^4 + 264*x^6 + 3720*x^8 +...
%e A172393 where A(x)^2 equals the g.f. of A172391:
%e A172393 A172391=[1,8,12,0,28,0,264,0,3720,0,63840,0,1232432,0,25731216,0,...].
%e A172393 Let G(x) = A(x*G(x)^2) = Sum_{n>=0} C(2n+2,n+1)/(n+2)*C(2n,n)*x^n:
%e A172393 G(x) = 1 + 2*2*x + 5*6*x^2 + 14*20*x^3 + 42*70*x^4 + 132*252*x^5 +...
%o A172393 (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))^(1/2),n)}
%Y A172393 Cf. A172391, A172392.
%K A172393 sign
%O A172393 0,2
%A A172393 _Paul D. Hanna_, Feb 05 2010