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%I #16 Mar 30 2012 18:37:33
%S 1,1,2,5,15,48,161,557,1975,7140,26223,97567,366979,1393104,5330500,
%T 20537289,79605148,310211668,1214618931,4776089725,18852665026,
%U 74676503735,296736116911,1182534483734,4725095364115,18926460148937,75981766859715,305674059089303,1232118979522167
%N G.f. satisfies: A(x) = Product_{n>=0} 1/( (1 - (x*A(x))^(5*n+1)) * (1 - (x*A(x))^(5*n+4)) ).
%C G.f. is an eigenfunction of the Rogers-Ramanujan identity described by A003114: Sum_{n>=0} x^(n^2)/(Product_{k=1..n} 1-x^k) = Product_{n>=0} 1/((1-x^(5*n+1))*(1-x^(5*n+4))).
%F G.f. satisfies:
%F (1) A(x) = Sum_{n>=0} x^(n^2)*A(x)^(n^2) / (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) / (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 A003114.
%F (4) A(x) = (1/x)*Series_Reversion(x/G(x)) where G(x) is the g.f. of A003114.
%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 48*x^5 + 161*x^6 + 557*x^7 +...
%e where the g.f. satisfies the Rogers-Ramanujan identity:
%e A(x) = 1/( (1 - x*A(x))*(1 - x^4*A(x)^4) * (1 - x^6*A(x)^6)*(1 - x^9*A(x)^9) * (1 - x^11*A(x)^11)*(1 - x^14*A(x)^14) * (1 - x^16*A(x)^16)*(1 - x^19*A(x)^19) *...);
%e A(x) = 1 + x*A(x)/(1-x*A(x)) + x^4*A(x)^4/((1-x*A(x))*(1-x^2*A(x)^2)) + x^9*A(x)^9/((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 A003114:
%e G(x) = 1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + 6*x^10 + 7*x^11 + 9*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)/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+1))*(1-(x*A)^(5*m+4)))+x*O(x^n)));polcoeff(A,n)}
%Y Cf. A203068, A003114.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 28 2011
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