OFFSET
0,3
COMMENTS
The g.f. of this sequence equals the numerator of George E. Andrews' expression for the cube of Ramanujan's continued fraction. See references given in A007325.
FORMULA
G.f.: Product_{n>=0} (1 - x^(n+1)) * (1 - x^(5*n+5)) / ( (1 - x^(5*n+2))^3 * (1 - x^(5*n+3))^3 ).
G.f.: Product_{n>=0} (1 - x^(5*n+5))^2 * (1 - x^(5*n+1))*(1 - x^(5*n+4)) / ( (1 - x^(5*n+2))^2*(1 - x^(5*n+3))^2 ).
G.f.: [ Sum_{n>=0} x^n/(1 - x^(5*n+3)) - x * Sum_{n>=0} x^(4*n)/(1 - x^(5*n+2)) ] * R(x), where R(q) is the expansion of Ramanujan's continued fraction (A007325).
EXAMPLE
G.f.: P(q) = 1 - q + 2*q^2 + 2*q^6 - 2*q^7 + 2*q^8 + q^9 + q^12 - 2*q^13 + 2*q^14 + 2*q^16 + 2*q^18 - 2*q^19 + 2*q^23 + 2*q^24 - 2*q^25 + 2*q^26 - q^28 + ...
Given the g.f. of this sequence,
P(q) = Sum_{n>=0} q^(2*n)/(1 - q^(5*n+2)) - q*Sum_{n>=0} q^(3*n)/(1 - q^(5*n+3))
and the g.f. of A340456,
Q(q) = Sum_{n>=0} q^n/(1 - q^(5*n+1)) - q^3*Sum_{n>=0} q^(4*n)/(1 - q^(5*n+4))
then
R(q)^3 = P(q)/Q(q) where
Q(q) = 1 + 2*q + 2*q^2 + q^3 + 2*q^4 + 2*q^5 + 2*q^6 + q^7 + 2*q^8 + 2*q^9 + 2*q^10 + 2*q^12 + 4*q^13 + 2*q^14 + q^16 + ...
R(q)^3 = 1 - 3*q + 6*q^2 - 7*q^3 + 3*q^4 + 6*q^5 - 17*q^6 + 24*q^7 - 21*q^8 + 6*q^9 + 21*q^10 - 54*q^11 + 77*q^12 - 72*q^13 + 24*q^14 + 64*q^15 + ...;
here, R(q) is the expansion of Ramanujan's continued fraction (A007325).
PROG
(PARI) {a(n) = my(A = prod(m=0, n\5+1, (1-x^(5*m+5) +x*O(x^n))^2 * (1-x^(5*m+1))*(1-x^(5*m+4)) / ( (1-x^(5*m+2))^2*(1-x^(5*m+3))^2 +x*O(x^n) ) )); polcoeff(A, n)}
for(n=0, 100, print1(a(n), ", "))
(PARI) {S(j, k, n) = sum(m=0, n, x^(j*m)/(1 - x^(5*m+k) +x*O(x^n)) ) }
{a(n) = polcoeff( S(2, 2, n) - x*S(3, 3, n), n)}
for(n=0, 100, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jan 20 2021
STATUS
approved