A256637 Expansion of psi(-q) * phi(-q^3)^2 / (q * psi(-q^3)^3) in powers of q where phi(), psi() are Ramanujan theta functions.

1, -1, 0, -2, 1, 0, 0, 2, 0, 2, 0, 0, 0, -4, 0, -4, -1, 0, 0, 6, 0, 8, 0, 0, 1, -10, 0, -12, 1, 0, 0, 16, 0, 18, 0, 0, -2, -24, 0, -28, -1, 0, 0, 36, 0, 40, 0, 0, 2, -52, 0, -58, 2, 0, 0, 74, 0, 84, 0, 0, -2, -104, 0, -116, -3, 0, 0, 144, 0, 160, 0, 0, 3

Offset

-1,4

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = -1..1000

Johannes Blümlein, Iterative Non-iterative Integrals in Quantum Field Theory, arXiv:1808.08128 [hep-th], 2018.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

Expansion of eta(q) * eta(q^3) * eta(q^4) * eta(q^6) / (eta(q^2) * eta(q^12)^3) in powers of q.

Euler transform of period 12 sequence [ -1, 0, -2, -1, -1, -2, -1, -1, -2, 0, -1, 0, ...].

a(3*n + 1) = a(4*n + 1) = 0. a(2*n) = - A139137(n). a(4*n - 1) = A256626(n).

a(6*n) = - A132002(n). a(6*n + 2) = -2 * A139135(n). a(12*n - 1) = A256636(n). a(12*n + 3) = A101195(n).

Example

G.f. = 1/q - 1 - 2*q^2 + q^3 + 2*q^6 + 2*q^8 - 4*q^12 - 4*q^14 - q^15 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[eta[q]* eta[q^3]*eta[q^4]*eta[q^6]/(eta[q^2]*eta[q^12]^3), {q, 0, n}]; Table[a[n], {n, -1, 100}] (* G. C. Greubel, Mar 14 2018 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^6 + A) / (eta(x^2 + A) * eta(x^12 + A)^3), n))};

Crossrefs

Cf. A101195, A139137, A256626, A256636.

Sequence in context: A040081 A066745 A239393 * A113063 A123477 A035225

Adjacent sequences: A256634 A256635 A256636 * A256638 A256639 A256640

Keyword

sign

Author

Michael Somos, Apr 06 2015

Status

approved