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A178324 G.f.: A(x) = Sum_{n>=0} x^n/[Sum_{k=0..n} C(n,k)^2*(-x)^k].

%I A178324 #12 Nov 09 2014 15:11:45
%S A178324 1,1,2,6,26,146,1024,8776,89670,1071174,14749528,231333872,4090080388,
%T A178324 80808196548,1770893277532,42769289395020,1131848169152498,
%U A178324 32655896028179970,1022576267852973448,34612454839065740688
%N A178324 G.f.: A(x) = Sum_{n>=0} x^n/[Sum_{k=0..n} C(n,k)^2*(-x)^k].
%C A178324 Compare the g.f. of this sequence to the identity:
%C A178324 (1-x)/(1-2*x) = Sum_{n>=0} x^n/[Sum_{k=0..n} C(n,k)*(-x)^k].
%F A178324 G.f.: Sum_{n>=0} x^n/((1+x)^(2n+1)*[Sum_{k>=0} C(n+k,k)^2*(-x)^k]).
%e A178324 G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 26*x^4 + 146*x^5 + 1024*x^6 +...
%e A178324 which equals the series:
%e A178324 A(x) = 1 + x/(1-x) + x^2/(1-4*x+x^2) + x^3/(1-9*x+9*x^2-x^3) + x^4/(1-16*x+36*x^2-16*x^3+x^4) + x^5/(1-25*x+100*x^2-100*x^3+25*x^4-x^5) +...
%e A178324 Also, the g.f. equals the series:
%e A178324 A(x) = 1/((1+x)*(1 - x + x^2 - x^3 +...))
%e A178324 + x/((1+x)^3*(1 - 2^2*x + 3^2*x^2 - 4^2*x^3 +...))
%e A178324 + x^2/((1+x)^5*(1 - 3^2*x + 6^2*x^2 - 10^2*x^3 +...))
%e A178324 + x^3/((1+x)^7*(1 - 4^2*x + 10^2*x^2 - 20^2*x^3 +...)) +...
%o A178324 (PARI) {a(n)=polcoeff(sum(m=0,n,x^m/sum(k=0,m,binomial(m,k)^2*(-x)^k+x*O(x^n))),n)}
%o A178324 for(n=0,25,print1(a(n),", "))
%o A178324 (PARI) {a(n)=polcoeff(sum(m=0, n, x^m/((1+x+x*O(x^n))^(2*m+1)*sum(k=0, n-m+1, binomial(m+k, k)^2*(-x)^k+x*O(x^n)))), n)}
%o A178324 for(n=0,25,print1(a(n),", "))
%K A178324 nonn
%O A178324 0,3
%A A178324 _Paul D. Hanna_, Dec 21 2010