A340453 G.f.: Product_{n>=0} (1 - x^(5*n+5))^2 / ( (1 - x^(5*n+1))*(1 - x^(5*n+4)) ).

1, 1, 1, 1, 2, 0, 1, 1, 2, 1, 1, 0, 2, 1, 0, 2, 2, 0, 1, 1, 2, 1, 1, -1, 2, 1, 2, 1, 2, 1, 0, 1, 1, 1, 1, 0, 2, 2, 1, 1, 2, -1, 1, 2, 2, 1, 0, 0, 3, 0, 1, 1, 2, 0, 1, 1, 2, 2, 1, 0, 2, 1, 0, 1, 2, 0, 1, 1, 0, 1, 2, 2, 2, 1, 2, 0, 2, -1, 0, 1, 2

Offset: 0

Author

Paul D. Hanna, Jan 16 2021

Keywords

• sign

Examples

			G.f.: P(q) = 1 + q + q^2 + q^3 + 2*q^4 + q^6 + q^7 + 2*q^8 + q^9 + q^10 + 2*q^12 + q^13 + 2*q^15 + 2*q^16 + q^18 + q^19 + 2*q^20 + ...
Given the g.f. of this sequence,
P(q) = Product_{n>=0} (1 - q^(5*n+5))^2 / ( (1 - q^(5*n+1))*(1 - q^(5*n+4)) ),
and the g.f. of A340454,
Q(q) = Product_{n>=0} (1 - q^(5*n+5))^2 / ( (1 - q^(5*n+2))*(1 - q^(5*n+3)) ),
then R(q) = P(q)/Q(q) where
Q(q) = 1 + q^2 + q^3 + q^4 - q^5 + 2*q^6 + q^8 + q^9 + q^10 + q^12 - q^13 + q^14 + 2*q^15 + q^16 + 2*q^18 - q^19 + q^20 + ...
and
R(q) = 1 + q - q^3 + q^5 + q^6 - q^7 - 2*q^8 + 2*q^10 + 2*q^11 - q^12 - 3*q^13 - q^14 + 3*q^15 + 3*q^16 - 2*q^17 - 5*q^18 - q^19 + 6*q^20 + ...;
here, R(q) is the expansion of Ramanujan's continued fraction (A007325).
		

Crossrefs

Cf. A007325, A340454, A340455, A340456.

Programs

PARI

{a(n) = my(A = prod(m=0,n, (1 - x^(5*m+5))^2 / ( (1 - x^(5*m+1))*(1 - x^(5*m+4)) +x*O(x^n)) )); polcoeff(A,n)}
for(n=0,80,print1(a(n),", "))

PARI

{a(n) = my(A = sum(m=0,n, x^(1*m)/(1 - x^(5*m+2) +x*O(x^n)) ) - x^2 * sum(m=0,n, x^(3*m)/(1 - x^(5*m+4) +x*O(x^n)) )); polcoeff(A,n)}
for(n=0,80,print1(a(n),", "))

PARI

{a(n) = my(A = sum(m=0,n, x^(2*m)/(1 - x^(5*m+1) +x*O(x^n)) ) - x^2 * sum(m=0,n, x^(4*m)/(1 - x^(5*m+3) +x*O(x^n)) )); polcoeff(A,n)}
for(n=0,80,print1(a(n),", "))

Formula

G.f.: Sum_{n>=0} x^(1*n)/(1 - x^(5*n+2)) - x^2 * Sum_{n>=0} x^(3*n)/(1 - x^(5*n+4)).

G.f.: Sum_{n>=0} x^(2*n)/(1 - x^(5*n+1)) - x^2 * Sum_{n>=0} x^(4*n)/(1 - x^(5*n+3)).

G.f.: Sum_{n>=0} x^(1*n)/(1 - x^(5*n+2)) - x^2 * Sum_{n>=0} x^(4*n)/(1 - x^(5*n+3)).

G.f.: Sum_{n>=0} x^(2*n)/(1 - x^(5*n+1)) - x^2 * Sum_{n>=0} x^(3*n)/(1 - x^(5*n+4)).

G.f.: [ Sum_{n>=0} x^n/(1 - x^(5*n+1)) - x^3 * Sum_{n>=0} x^(4*n)/(1 - x^(5*n+4)) ] * R(x), where R(q) is the expansion of Ramanujan's continued fraction (A007325).