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A192627 G.f. satisfies: A(x) = Product_{n>=0} (1 + x*(x+x^2)^n)/(1 - x*(x+x^2)^n).

%I A192627 #8 Aug 14 2025 13:52:36
%S A192627 1,2,4,10,22,48,104,222,466,966,1988,4060,8222,16528,33024,65620,
%T A192627 129698,255096,499508,974032,1891866,3661034,7060324,13572010,
%U A192627 26009822,49701946,94714606,180022550,341316642,645594510,1218377230,2294387492,4311757732
%N A192627 G.f. satisfies: A(x) = Product_{n>=0} (1 + x*(x+x^2)^n)/(1 - x*(x+x^2)^n).
%C A192627 Related q-series identity due to Heine:
%C A192627 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 A192627 here q=x+x^2, x=x, y=1, z=0.
%F A192627 G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n*(x+x^2)^(n*(n-1)/2) * Product_{k=0..n-1} (1+(x+x^2)^k)/((1-x*(x+x^2)^k)*(1-(x+x^2)^(k+1)) due to the Heine identity.
%F A192627 Self-convolution equals A192626.
%e A192627 G.f.: A(x) = 1 + 2*x + 4*x^2 + 10*x^3 + 22*x^4 + 48*x^5 + 104*x^6 +...
%e A192627 where the g.f. equals the product:
%e A192627 A(x) = (1+x)/(1-x) * (1+x^2*(1+x))/(1-x^2*(1+x)) * (1+x^3*(1+x)^2)/(1-x^3*(1+x)^2) * (1+x^4*(1+x)^3)/(1-x^4*(1+x)^3) *...
%e A192627 which is also equal to the sum:
%e A192627 A(x) = 1 + 2*x/((1-x)*(1-x*(1+x))) + 2*x^3*(1+x)*(1+x*(1+x))/((1-x)*(1-x*(1+x))*(1-x^2*(1+x))*(1-x^2*(1+x)^2)) + 2*x^6*(1+x)*(1+x*(1+x))*(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 A192627 (PARI) {a(n)=local(A=1+x);A=prod(k=0,n+1,(1+x*(x+x^2)^k)/(1-x*(x+x^2+x*O(x^n))^k));polcoeff(A,n)}
%o A192627 (PARI) {a(n)=local(A=1+x);A=1+sum(m=1,n,x^m*(x+x^2)^(m*(m-1)/2)*prod(k=0,m-1,(1+(x+x^2)^k)/((1-x*(x+x^2)^k +x*O(x^n))*(1-(x+x^2)^(k+1)))));polcoeff(A,n)}
%Y A192627 Cf. A192626, A192623.
%K A192627 nonn
%O A192627 0,2
%A A192627 _Paul D. Hanna_, Jul 16 2011