A182031 Expansion of q^(-5/24) * (eta(q^3) * eta(q^6))^3 / (eta(q) * eta(q^2))^4 in powers of q.

1, 4, 18, 53, 163, 414, 1059, 2431, 5553, 11844, 25013, 50391, 100362, 193136, 367371, 680705, 1247247, 2238408, 3975218, 6941384, 12003156, 20465599, 34581525, 57737205, 95601892, 156665029, 254777220, 410580026, 657015874

Offset

0,2

Comments

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

References

H.-C. Chan, On the Andrews-Schur proof of the Rogers-Ramanujan identities, Ramanujan J. 23 (2010), no. 1-3, 417-431. see p. 430 Theorem 7.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (psi(x^3) * phi(-x^3))^3 / (psi(x) * phi(-x))^4 in powers of x where phi(), psi() are Ramanujan theta functions.

Euler transform of period 6 sequence [ 4, 8, 1, 8, 4, 2, ...].

A002513(3*n + 2) = 3 * a(n).

Example

1 + 4*x + 18*x^2 + 53*x^3 + 163*x^4 + 414*x^5 + 1059*x^6 + 2431*x^7 + ...
q^5 + 4*q^13 + 18*q^21 + 53*q^29 + 163*q^37 + 414*q^45 + 1059*q^53 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-5/8)*(eta[q^3]*eta[q^6])^3/(eta[q]*eta[q^2])^4, {q, 0, 100}], q] (* G. C. Greubel, Apr 16 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^6 + A))^3 / (eta(x + A) * eta(x^2 + A))^4, n))}
(PARI) q='q+O('q^99); Vec((eta(q^3)*eta(q^6))^3/(eta(q)*eta(q^2))^4) \\ Altug Alkan, Apr 16 2018

Crossrefs

Cf. A002513.

Sequence in context: A297945 A320544 A020644 * A212250 A229788 A242206

Adjacent sequences: A182028 A182029 A182030 * A182032 A182033 A182034

Keyword

nonn

Author

Michael Somos, Apr 07 2012

Status

approved