A226034 Expansion of f(-x)^6 / (chi(x) * phi(-x)^6) in powers of x where phi(), chi(), f() are Ramanujan theta functions.
1, 11, 73, 368, 1552, 5755, 19337, 60054, 174801, 481760, 1266992, 3198963, 7791921, 18382187, 42139440, 94126547, 205343040, 438390320, 917501570, 1885269635, 3808353889, 7571955531, 14833349529, 28657374307, 54646711136, 102932171227, 191644299945
Offset: 0
Keywords
Examples
1 + 11*x + 73*x^2 + 368*x^3 + 1552*x^4 + 5755*x^5 + 19337*x^6 + 60054*x^7 + ... q^19 + 11*q^43 + 73*q^67 + 368*q^91 + 1552*q^115 + 5755*q^139 + 19337*q^163 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Crossrefs
Cf. A001935.
Programs
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Mathematica
nmax=60; CoefficientList[Series[Product[(1+x^k)^4 * (1-x^(3*k))^6 * (1-x^(4*k)) / (1-x^k)^7,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 14 2015 *) eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[q^(-19/24)* eta[q^2]^4*eta[q^3]^6*eta[q^4]/eta[q]^11, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 15 2018 *) -
PARI
{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^3 + A)^6 * eta(x^4 + A) / eta(x + A)^11, n))}
Formula
Expansion of q^(-19/24) * eta(q^2)^4 * eta(q^3)^6 * eta(q^4) / eta(q)^11 in powers of q.
a(n) = 1/12 * A001935(9*n + 7).
a(n) ~ exp(3*Pi*sqrt(n/2)) / (2^(19/4) * 3^(5/2) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015
Comments