A143895 Expansion of (chi(q)^4 / chi(-q))^2 in powers of q where chi() is a Ramanujan theta function.

1, 10, 47, 150, 403, 1002, 2316, 5004, 10309, 20456, 39240, 73102, 132779, 235868, 410785, 702630, 1182342, 1960418, 3206675, 5179670, 8270086, 13062994, 20427293, 31644200, 48589970, 73994118, 111802523, 167685238, 249745021, 369499928

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^(1/4) * (eta(q^2)^9 / (eta(q)^5 * eta(q^4)^4))^2 in powers of q.

Euler transform of period 4 sequence [ 10, -8, 10, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A143894.

G.f.: (Product_{k>0} (1 + x^k)^5 / (1 + x^(2*k))^4)^2.

a(n) ~ exp(sqrt(2*n)*Pi) / (2^(9/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015

Example

G.f. = 1 + 10*x + 47*x^2 + 150*x^3 + 403*x^4 + 1002*x^5 + 2316*x^6 + 5004*x^7 + ...
G.f. = 1/q + 10*q^3 + 47*q^7 + 150*q^11 + 403*q^15 + 1002*q^19 + 2316*q^23 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^9 / (QPochhammer[ x]^5 QPochhammer[ x^4]^4))^2, {x, 0, n}]; (* Michael Somos, Apr 26 2015 *)
nmax = 40; CoefficientList[Series[Product[((1 + x^k)^5 / (1 + x^(2*k))^4)^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)^5 * eta(x^4 + A)^4))^2, n))};

Crossrefs

Cf. A143894.

Sequence in context: A000832 A242462 A253477 * A281767 A323799 A213575

Adjacent sequences: A143892 A143893 A143894 * A143896 A143897 A143898

Keyword

nonn

Author

Michael Somos, Sep 04 2008

Status

approved