A032442 Expansion of 1 / Product_{k >= 1} (1-q^k)^2*(1-q^(11k))^2.
1, 2, 5, 10, 20, 36, 65, 110, 185, 300, 481, 754, 1169, 1780, 2685, 3996, 5894, 8600, 12450, 17860, 25442, 35964, 50519, 70490, 97800, 134892, 185099, 252664, 343280, 464200, 625033, 837998, 1119114, 1488720, 1973210, 2606028, 3430238, 4500224, 5885540
Offset: 0
Keywords
Examples
G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 36*x^5 + 65*x^6 + 110*x^7 + ... G.f. = 1/q + 2 + 5*q + 10*q^2 + 20*q^3 + 36*q^4 + 65*q^5 + 110*q^6 + ...
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], 2015-2016.
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
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Mathematica
a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^11])^-2, {x, 0, n}]; (* Michael Somos, Apr 21 2015 *) nmax=60; CoefficientList[Series[Product[1/((1-x^k)^2 * (1-x^(11*k))^2),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *) -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^11 + A))^-2, n))}; /* Michael Somos, Apr 21 2015 */
Formula
Expansion of 1 / (f(-x) * f(-x^11))^2 in powers of x where f() is a Ramanujan theta function. - Michael Somos, Apr 21 2015
Expansion of q / eta(q)^2 * eta(q^11)^2 in powers of q. - Michael Somos, Apr 21 2015
Euler transform of period 11 sequence [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, ...]. - Michael Somos, Apr 21 2015
Given g.f. A(x), then B(q) = A(q)/q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2 * (w^2 + 16*v^2) - v^2 * (v + 4*u) * (w + 4*u). - Michael Somos, Apr 21 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11^-1 (t/i)^-2 f(t) where q = exp(2 Pi i t). - Michael Somos, Apr 21 2015
G.f.: (Product_{k > 0} (1 - x^k)^2 * (1 - x^(11*k)))^-2.
a(n) ~ exp(4*Pi*sqrt(n/11)) / (sqrt(2) * 11^(1/4) * n^(7/4)). - Vaclav Kotesovec, Oct 13 2015
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