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

Author

Michael Somos, Sep 04 2008

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

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

Crossrefs

Cf. A022571, A052241, A143895.

Programs

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(2*n) = A052241(n). a(2*n + 1) = 8 * A022571(n).

a(n) ~ exp(sqrt(n)*Pi) / (sqrt(2) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015