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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.

%I A187100 #40 Feb 16 2025 08:33:14
%S A187100 1,3,7,15,30,57,104,183,313,522,852,1365,2150,3336,5106,7719,11538,
%T A187100 17067,25004,36306,52280,74700,105960,149277,208951,290706,402127,
%U A187100 553224,757158,1031166,1397744,1886151,2534316,3391254,4520112,6002007,7940846
%N 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.
%C A187100 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).
%H A187100 G. C. Greubel, <a href="/A187100/b187100.txt">Table of n, a(n) for n = 1..1000</a>
%H A187100 Kevin Acres, David Broadhurst, <a href="https://arxiv.org/abs/1810.07478">Eta quotients and Rademacher sums</a>, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.
%H A187100 Johannes Blümlein, <a href="https://arxiv.org/abs/1808.08128">Iterative Non-iterative Integrals in Quantum Field Theory</a>, arXiv:1808.08128 [hep-th], 2018.
%H A187100 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A187100 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.
%F A187100 Euler transform of period 12 sequence [ 3, 1, 2, 2, 3, 2, 3, 2, 2, 1, 3, 0, ...].
%F A187100 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).
%F A187100 Convolution inverse of A187130. A186924(n) = 4 * a(n) unless n=0.
%F A187100 a(n) ~ exp(2*Pi*sqrt(n/3)) / (8 * 3^(5/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 10 2015
%e A187100 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 + ...
%t A187100 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 *)
%t A187100 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 *)
%t A187100 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 *)
%o A187100 (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))}
%Y A187100 Cf. A186924, A187130, A121373, A000122, A010054, A000700.
%K A187100 nonn
%O A187100 1,2
%A A187100 _Michael Somos_, Mar 05 2011