OFFSET
0,2
COMMENTS
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of chi(q)^12 in powers of q where chi() is a Ramanujan theta function.
Expansion of q^(1/2) * (eta(q^2)^2 / (eta(q) * eta(q^4)))^12 in powers of q.
G.f.: Product_{k>0} (1 + (-x)^k)^-12 = Product_{k>0} (1 + x^(2*k - 1))^-12.
G.f.: T(0), where T(k) = 1 - 1/(1 - 1/(1 - 1/(1+(x)^(2*k+1))^12/T(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 06 2013
a(n) ~ exp(Pi*sqrt(2*n)) / (2^(5/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
G.f.: exp(12*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 07 2018
EXAMPLE
1 + 12*x + 66*x^2 + 232*x^3 + 639*x^4 + 1596*x^5 + 3774*x^6 + 8328*x^7 + ...
T8B = 1/q + 12*q + 66*q^3 + 232*q^5 + 639*q^7 + 1596*q^9 + 3774*q^11 + ...
MATHEMATICA
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ ((1 - m) m / 16 / q)^(1/2), {q, 0, n}]] (* Michael Somos, Jul 22 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, 1, n, 2}]^-12, {x, 0, n}] (* Michael Somos, Jul 22 2011 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^12, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)
QP = QPochhammer; s = (QP[q^2]^2/(QP[q]*QP[q^4]))^12 + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 16 2015, adapted from PARI *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A)))^12, n))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 28 2005
STATUS
approved