A340454 G.f.: Product_{n>=0} (1 - x^(5*n+5))^2 / ( (1 - x^(5*n+2))*(1 - x^(5*n+3)) ).

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

Offset

0,7

Links

Table of n, a(n) for n=0..80.

Formula

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

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

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

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

G.f.: [ Sum_{n>=0} x^(2*n)/(1 - x^(5*n+2)) - x * Sum_{n>=0} x^(3*n)/(1 - x^(5*n+3)) ] / R(x), where R(q) is the expansion of Ramanujan's continued fraction (A007325).

Example

G.f.: 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 + ...
Given the g.f. of this sequence,
Q(q) = Product_{n>=0} (1 - q^(5*n+5))^2 / ( (1 - q^(5*n+2))*(1 - q^(5*n+3)) ),
and the g.f. of A340453,
P(q) = Product_{n>=0} (1 - q^(5*n+5))^2 / ( (1 - q^(5*n+1))*(1 - q^(5*n+4)) ),
then R(q) = P(q)/Q(q) where
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 + ...
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).

Prog

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

Crossrefs

Cf. A007325, A340453, A340455, A340456.

Sequence in context: A074272 A328372 A341022 * A342682 A105553 A342026

Adjacent sequences: A340451 A340452 A340453 * A340455 A340456 A340457

Keyword

sign

Author

Paul D. Hanna, Jan 16 2021

Status

approved