A340456 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)).

1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 0, 2, 4, 2, 0, 1, 2, 2, 2, 2, 2, 2, 0, 3, 2, 2, 0, 2, 2, 2, 2, 2, 2, 0, 2, 2, 4, 2, -1, 2, 2, 2, 0, 2, 2, 4, 0, 2, 4, 2, 1, 0, 0, 2, 2, 2, 4, 2, 0, 2, 2, 2, 0, 2, 2, 2, 2, 4, 2, 0, 0, 1, 4, 2, 0, 2, 2, 2, 2, 2, 0, 2, 2, 2, 4

Offset

0,2

Comments

The g.f. of this sequence equals the denominator 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..85.

Formula

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

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

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)) ] / R(x), where R(q) is the expansion of Ramanujan's continued fraction (A007325).

Example

G.f.: 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 + 2*q^17 + 2*q^18 + 2*q^19 + 2*q^20 + ...
Given the g.f. of this sequence
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))
and the g.f. of A340455,
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))
then
R(q)^3 = P(q)/Q(q) where
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 + ...
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+2))*(1-x^(5*m+3)) / ( (1-x^(5*m+1))^2*(1-x^(5*m+4))^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(1, 1, n) - x^3*S(4, 4, n), n)}
for(n=0, 100, print1(a(n), ", "))

Crossrefs

Cf. A007325, A055102, A340455, A340453, A340454.

Sequence in context: A078377 A291454 A105697 * A080757 A037196 A368594

Adjacent sequences: A340453 A340454 A340455 * A340457 A340458 A340459

Keyword

sign

Author

Paul D. Hanna, Jan 20 2021

Status

approved