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

0,2

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 = 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

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

Example

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 + ...

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 *)

Prog

(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))}

Crossrefs

Cf. A001935.

Sequence in context: A197308 A142015 A123039 * A217946 A163775 A092244

Adjacent sequences: A226031 A226032 A226033 * A226035 A226036 A226037

Keyword

nonn

Author

Michael Somos, May 28 2013

Status

approved