OFFSET
0,5
COMMENTS
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))).
FORMULA
G.f. satisfies:
(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).
(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))).
(3) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A003106.
(4) A(x) = (1/x)*Series_Reversion(x/G(x)) where G(x) is the g.f. of A003106.
EXAMPLE
G.f.: A(x) = 1 + x^2 + x^3 + 3*x^4 + 6*x^5 + 16*x^6 + 37*x^7 + 101*x^8 +...
where the g.f. satisfies the Rogers-Ramanujan identity:
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) *...);
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)) +...
Also, A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A003106:
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 +...
PROG
(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)}
(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)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 28 2011
STATUS
approved