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

0,2

Comments

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

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

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

Example

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 + ...

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 *)

Prog

(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))};

Crossrefs

Cf. A022571, A052241, A143895.

Sequence in context: A031307 A122854 A138502 * A126176 A240754 A074238

Adjacent sequences: A143891 A143892 A143893 * A143895 A143896 A143897

Keyword

nonn

Author

Michael Somos, Sep 04 2008

Status

approved