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A192626 G.f. satisfies: A(x) = Product_{n>=0} (1 + x*(x+x^2)^n)^2/(1 - x*(x+x^2)^n)^2.

%I A192626 #8 Aug 14 2025 13:52:40
%S A192626 1,4,12,36,100,264,676,1684,4096,9764,22888,52872,120540,271600,
%T A192626 605556,1337320,2927720,6358432,13707916,29351536,62450468,132090356,
%U A192626 277845120,581405140,1210688864,2509483020,5178969644,10644112012,21790816340,44444609044
%N A192626 G.f. satisfies: A(x) = Product_{n>=0} (1 + x*(x+x^2)^n)^2/(1 - x*(x+x^2)^n)^2.
%C A192626 Related q-series identity due to Heine:
%C A192626 1 + Sum_{n>=1} x^n*Product_{k=0..n-1} (y+q^k)*(z+q^k)/((1-x*q^k)*(1-q^(k+1)) = Product_{n>=0} (1+x*y*q^n)*(1+x*z*q^n)/((1-x*q^n)*(1-x*y*z*q^n));
%C A192626 here q=x+x^2, x=x, y=z=1.
%F A192626 G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n*Product_{k=0..n-1} (1 + x^k*(1+x)^k)^2/((1 - x^(k+1)*(1+x)^k)*(1 - x^(k+1)*(1+x)^(k+1))) due to the Heine identity.
%F A192626 Self-convolution of A192627.
%e A192626 G.f.: A(x) = 1 + 4*x + 12*x^2 + 36*x^3 + 100*x^4 + 264*x^5 + 676*x^6 +...
%e A192626 where the g.f. equals the product:
%e A192626 A(x) = (1+x)^2/(1-x)^2 * (1+x^2*(1+x))^2/(1-x^2*(1+x))^2 * (1+x^3*(1+x)^2)^2/(1-x^3*(1+x)^2)^2 * (1+x^4*(1+x)^3)^2/(1-x^4*(1+x)^3)^2 *...
%e A192626 which is also equal to the sum:
%e A192626 A(x) = 1 + 4*x/((1-x)*(1-x*(1+x))) + 4*x^2*(1+x*(1+x))^2/((1-x)*(1-x*(1+x))*(1-x^2*(1+x))*(1-x^2*(1+x)^2)) + 4*x^3*(1+x*(1+x))^2*(1 + x^2*(1+x)^2)^2/((1-x)*(1-x*(1+x))*(1-x^2*(1+x))*(1-x^2*(1+x)^2)*(1-x^3*(1+x)^2)*(1-x^3*(1+x)^3)) +...
%o A192626 (PARI) {a(n)=local(A=1+x);A=prod(k=0,n+1,(1+x*(x+x^2)^k)^2/(1-x*(x+x^2+x*O(x^n))^k)^2);polcoeff(A,n)}
%o A192626 (PARI) {a(n)=local(A=1+x);A=1+sum(m=1,n,x^m*prod(k=0,m-1,(1+(x+x^2)^k)^2/((1-x*(x+x^2)^k +x*O(x^n))*(1-(x+x^2)^(k+1)))));polcoeff(A,n)}
%Y A192626 Cf. A192627, A192622.
%K A192626 nonn
%O A192626 0,2
%A A192626 _Paul D. Hanna_, Jul 16 2011