A340455 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)).

1, -1, 2, 0, 0, 0, 2, -2, 2, 1, 0, 0, 1, -2, 2, 0, 2, 0, 2, -2, 0, 0, 0, 2, 2, -2, 2, 0, -1, 0, 4, -2, 2, -1, 0, 0, 0, 0, 2, 0, 2, 0, 2, -2, 2, 0, -2, 0, 2, -2, 2, 2, 0, 0, 2, -2, 2, 1, 2, -2, 0, -2, 2, 0, 1, 2, 2, -2, 0, 0, 0, 0, 2, -2, 4

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.

Links

Table of n, a(n) for n=0..74.

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

Cf. A007325, A055102, A340456, A340453, A340454.

Sequence in context: A291957 A143063 A210703 * A275964 A284272 A175070

Adjacent sequences: A340452 A340453 A340454 * A340456 A340457 A340458

Keyword

sign

Author

Paul D. Hanna, Jan 20 2021

Status

approved