A145703 Expansion of chi(x) / chi(-x^10) in powers of x where chi() is a Ramanujan theta function.
1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 6, 6, 7, 8, 10, 11, 11, 13, 15, 17, 18, 20, 23, 25, 29, 32, 34, 39, 42, 47, 52, 56, 62, 68, 77, 83, 89, 99, 108, 119, 129, 139, 154, 167, 183, 199, 214, 234, 253, 276, 299, 322, 350, 378, 413, 445, 476, 518, 559
Offset: 0
Keywords
Examples
G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + 3*x^10 + ... G.f. = q^3 + q^11 + q^27 + q^35 + q^43 + q^51 + q^59 + 2*q^67 + 2*q^75 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
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Mathematica
nmax = 100; CoefficientList[Series[Product[(1 + x^(2*k-1)) / (1 - x^(20*k-10)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 30 2015 *) a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^10, x^10], {x, 0, n}]; (* Michael Somos, Sep 06 2015 *) -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^20 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^10 + A)), n))};
Formula
Expansion of q^(-3/8) * eta(q^2)^2 * eta(q^20) / (eta(q) * eta(q^4) * eta(q^10) ) in powers of q.
Euler transform of period 20 sequence [ 1, -1, 1, 0, 1, -1, 1, 0, 1, 0, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (640 t)) = 2^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A145702.
G.f.: Product_{k>0} (1 + x^(2*k - 1)) / (1 - x^(20*k - 10)).
a(n) ~ exp(Pi*sqrt(n/5)) / (4*5^(1/4)*n^(3/4)). - Vaclav Kotesovec, Aug 30 2015
Comments