A187100 Expansion of q * (psi(-q^3) * psi(q^6)) / (psi(-q) * phi(-q)) in powers of q where phi(), psi() are Ramanujan theta functions.

1, 3, 7, 15, 30, 57, 104, 183, 313, 522, 852, 1365, 2150, 3336, 5106, 7719, 11538, 17067, 25004, 36306, 52280, 74700, 105960, 149277, 208951, 290706, 402127, 553224, 757158, 1031166, 1397744, 1886151, 2534316, 3391254, 4520112, 6002007, 7940846

Offset: 1

Author

Michael Somos, Mar 05 2011

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) := Product_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k>=0} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).

Examples

			q + 3*q^2 + 7*q^3 + 15*q^4 + 30*q^5 + 57*q^6 + 104*q^7 + 183*q^8 + 313*q^9 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.
• Johannes Blümlein, Iterative Non-iterative Integrals in Quantum Field Theory, arXiv:1808.08128 [hep-th], 2018.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A186924, A187130, A121373, A000122, A010054, A000700.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[(1-x^(2*k))^2 * (1-x^(3*k)) * (1-x^(12*k))^3 / ((1-x^k)^3 * (1-x^(4*k)) * (1-x^(6*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 10 2015 *)
a[n_] := SeriesCoefficient[(EllipticTheta[2, 0, I*q^(3/2)]* EllipticTheta[2, 0, q^3])/(2*EllipticTheta[2, 0, I*q^(1/2)]* EllipticTheta[3, 0, -q]), {q, 0, n}]; Table[a[n], {n, 50}] (* G. C. Greubel, Nov 27 2017 *)
eta[q_] := q^(1/24)*QPochhammer[q]; A:= eta[q^2]^2*eta[q^3]*eta[q^12]^3/ (eta[q]^3*eta[q^4]*eta[q^6]^2); a:=CoefficientList[Series[A/q, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jul 01 2018 *)

PARI

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

Formula

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

Euler transform of period 12 sequence [ 3, 1, 2, 2, 3, 2, 3, 2, 2, 1, 3, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (1/12) * 1/f(t) where q = exp(2 Pi i t).

Convolution inverse of A187130. A186924(n) = 4 * a(n) unless n=0.

a(n) ~ exp(2*Pi*sqrt(n/3)) / (8 * 3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Sep 10 2015