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%I #13 Mar 30 2012 18:37:33
%S 1,0,1,1,3,6,16,37,101,252,691,1819,5014,13630,37915,105125,295229,
%T 829714,2350106,6671030,19035055,54452982,156358967,450089260,
%U 1299394185,3759776618,10904685696,31690917170,92283005769,269201392276,786639839881,2302254813072,6748059023185,19806420012004
%N G.f. satisfies: A(x) = Product_{n>=0} 1/( (1 - (x*A(x))^(5*n+2)) * (1 - (x*A(x))^(5*n+3)) ).
%C G.f. is an eigenfunction of the Rogers-Ramanujan identity described by A003106: Sum_{n>=0} x^(n^2+n)/(Product_{k=1..n} 1-x^k) = Product_{n>=0} 1/((1-x^(5*n+2))*(1-x^(5*n+3))).
%F G.f. satisfies:
%F (1) A(x) = Sum_{n>=0} x^(n^2+n)*A(x)^(n^2+n) / (Product_{k=1..n} 1 - x^k*A(x)^k).
%F (2) A(x) = ((Product_{n>0} 1 + (x*A(x))^(2*n)) * (Sum_{n>=0} (x*A(x))^(n^2+2*n) / (Product_{k=1..n} 1 - (x*A(x))^(4*k))).
%F (3) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A003106.
%F (4) A(x) = (1/x)*Series_Reversion(x/G(x)) where G(x) is the g.f. of A003106.
%e G.f.: A(x) = 1 + x^2 + x^3 + 3*x^4 + 6*x^5 + 16*x^6 + 37*x^7 + 101*x^8 +...
%e where the g.f. satisfies the Rogers-Ramanujan identity:
%e A(x) = 1/( (1 - x^2*A(x)^2)*(1 - x^3*A(x)^3) * (1 - x^7*A(x)^7)*(1 - x^8*A(x)^8) * (1 - x^12*A(x)^12)*(1 - x^13*A(x)^13) * (1 - x^17*A(x)^17)*(1 - x^18*A(x)^18) *...);
%e A(x) = 1 + x^2*A(x)^2/(1-x*A(x)) + x^6*A(x)^6/((1-x*A(x))*(1-x^2*A(x)^2)) + x^12*A(x)^12/((1-x*A(x))*(1-x^2*A(x)^2)*(1-x^3*A(x)^3)) +...
%e Also, A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A003106:
%e G(x) = 1 + x^2 + x^3 + x^4 + x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^9 + 4*x^10 + 4*x^11 + 6*x^12 +...
%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+sum(m=1,sqrtint(n+1),(x*A)^(m^2+m)/prod(k=1,m,1-(x*A)^k+x*O(x^n))));polcoeff(A,n)}
%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=prod(m=0,n,1/((1-(x*A)^(5*m+2))*(1-(x*A)^(5*m+3)))+x*O(x^n)));polcoeff(A,n)}
%Y Cf. A203067, A003106.
%K nonn
%O 0,5
%A _Paul D. Hanna_, Dec 28 2011
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