A095813 Expansion of q * chi(-q) / chi(-q^5)^5 in powers of q where chi() is a Ramanujan theta function.

1, -1, 0, -1, 1, 4, -4, -1, -3, 3, 12, -12, -2, -8, 8, 31, -30, -5, -20, 19, 72, -68, -12, -44, 41, 154, -144, -24, -90, 84, 312, -289, -48, -178, 164, 603, -554, -92, -336, 307, 1122, -1024, -168, -612, 557, 2024, -1836, -300, -1087, 983, 3552, -3206, -522, -1880, 1692, 6088, -5472, -886, -3180

Offset

1,6

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (1 - (phi(-q) / phi(-q^5))^2) / 4 in powers of q where phi() is a Ramanujan theta function.

Expansion of (eta(q) * eta(q^10)^5) / (eta(q^2) * eta(q^5)^5) in powers of q.

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v + 2*u*v + 4*u*v^2.

G.f. A(x) satisfies A(x^2) = -A(x) * A(-x).

Euler transform of period 10 sequence [ -1, 0, -1, 0, 4, 0, -1, 0, -1, 0, ...].

G.f.: x * (Product_{k>=1} ((1 - x^k) * (1-x^(10*k))^5) / ((1 - x^(2*k)) * (1 - x^(5*k))^5)).

a(n) = A138522(n) unless n = 0. Convolution inverse is A132980.

Empirical: Sum_{n>=1} a(n)/exp(Pi*n) = -13/8 - (5/8)*sqrt(5) + (5/8)*sqrt(10 + 6*sqrt(5)). - Simon Plouffe, Mar 01 2021

Example

q - q^2 - q^4 + q^5 + 4*q^6 - 4*q^7 - q^8 - 3*q^9 + 3*q^10 + 12*q^11 + ...

Prog

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^10 + A)^5 / (eta(x^2 + A) * eta(x^5 + A)^5), n))}

Crossrefs

Cf. A132980, A138522.

Sequence in context: A362329 A292434 A138522 * A010656 A350674 A321591

Adjacent sequences: A095810 A095811 A095812 * A095814 A095815 A095816

Keyword

sign

Author

Michael Somos, Jun 07 2004

Status

approved