A143894 Expansion of (chi(q)^5 * chi(-q))^2 in powers of q where chi() is a Ramanujan theta function.
1, 8, 26, 48, 79, 168, 326, 496, 755, 1296, 2106, 3072, 4460, 6840, 10284, 14448, 20165, 29184, 41640, 56880, 77352, 107472, 147902, 197616, 263019, 354888, 475516, 624048, 816065, 1076736, 1413142, 1826416, 2353446, 3050400, 3936754, 5022720
Offset: 0
Keywords
Examples
G.f. = 1 + 8*x + 26*x^2 + 48*x^3 + 79*x^4 + 168*x^5 + 326*x^6 + 496*x^7 + ... G.f. = 1/q + 8*q + 26*q^3 + 48*q^5 + 79*q^7 + 168*q^9 + 326*q^11 + 496*q^13 + ...
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
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^9 / (QPochhammer[ x]^4 QPochhammer[ x^4]^5))^2, {x, 0, n}]; (* Michael Somos, Apr 26 2015 *) nmax = 40; CoefficientList[Series[Product[((1 + x^k)^4 / (1 + x^(2*k))^5)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *) -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^9 / (eta(x + A)^4 * eta(x^4 + A)^5))^2, n))};
Formula
Expansion of q^(1/2) * (eta(q^2)^9 / (eta(q)^4 * eta(q^4)^5))^2 in powers of q.
Euler transform of period 4 sequence [ 8, -10, 8, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A143895.
G.f.: (Product_{k>0} (1 + x^k)^4 / (1 + x^(2*k))^5)^2.
a(n) ~ exp(sqrt(n)*Pi) / (sqrt(2) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015
Comments