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A199409 G.f. satisfies: A(x) = Sum_{n>=0} A(x)^n * x^(n^2) * (1 - x^(2*n+1))/(1 - x). 2

%I #12 Mar 30 2012 18:37:32

%S 1,1,2,4,8,17,37,82,184,417,954,2200,5109,11937,28040,66179,156857,

%T 373205,891034,2134072,5125944,12344835,29802478,72109852,174839832,

%U 424742526,1033697149,2519947080,6152807700,15045156972,36840289213,90326900587,221741403579,544982530105

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

%F Define f(z,q) = Sum_{n>=0} z^n * q^(n^2) then g.f. A(q) satisfies:

%F A(q) = (f(A(q),q) - q*f(q^2*A(q),q))/(1-q).

%e G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 17*x^5 + 37*x^6 + 82*x^7 +...

%e where the g.f. A(x) satisfies the equivalent expressions:

%e A(x) = 1 + A(x)*x*(1-x^3)/(1-x) + A(x)^2*x^4*(1-x^5)/(1-x) + A(x)^3*x^9*(1-x^7)/(1-x) + A(x)^4*x^16*(1-x^9)/(1-x) +...

%e A(x) = 1 + A(x)*(x + x^2 + x^3) + A(x)^2*(x^4 + x^5 + x^6 + x^7 + x^8) + A(x)^3*(x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15) +...

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,sqrtint(n+1),A^m*x^(m^2)*(1-x^(2*m+1))/(1-x))+x*O(x^n));polcoeff(A,n)}

%Y Cf. A199410.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 06 2011

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