A204372 Expansion of phi(x)^2 * (5 * phi(-x)^8 + 64 * x * psi(-x)^8) in powers of x where phi(), psi() are Ramanujan theta functions.

5, 4, 4, -320, 4, 2504, -320, -9600, 4, 25924, 2504, -58560, -320, 114248, -9600, -200320, 4, 334088, 25924, -521280, 2504, 768000, -58560, -1119360, -320, 1565004, 114248, -2099840, -9600, 2829128, -200320, -3694080, 4, 4684800

Offset

0,1

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..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of eta(q)^4 * eta(q^2)^2 * (5 * eta(q)^8 / eta(q^4)^4 + 64 * q * eta(q^4)^4 ) in powers of q.

G.f. is a period 1 Fourier series which satisfies f(-1 / (4*t)) = 2048 (t/i)^5 g(t) where q = exp(2*Pi*i*t) and g(t) is the g.f. for A050468.

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

a(n) = 4 * A050456(n) if n>0.

Example

G.f. = 5 + 4*x + 4*x^2 - 320*x^3 + 4*x^4 + 2504*x^5 - 320*x^6 - 9600*x^7 + 4*x^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^2 (5 EllipticTheta[ 4, 0, q]^8 + 4 EllipticTheta[ 2, Pi/4, q^(1/2)]^8), {q, 0, n}]; (* Michael Somos, May 03 2015 *)
a[ n_] := If[ n < 1, 5 Boole[n == 0], 4 DivisorSum[ n, #^4 KroneckerSymbol[ -4, #] &]]; (* Michael Somos, May 04 2015 *)

Prog

(PARI) {a(n) = if( n<1, 5 * (n==0), 4 * sumdiv( n, d, d^4 * kronecker( -4, d)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^2 + A)^2 * (5 * eta(x + A)^8 / eta(x^4 + A)^4 + 64 * x * eta(x^4 + A)^4 ), n))};
(Magma) A := Basis( ModularForms( Gamma1(4), 5), 34); 5*A[1] + 4*A[2] + 4*A[3]; /* Michael Somos, May 04 2015 */

Crossrefs

Cf. A050456, A050468.

Sequence in context: A071419 A291069 A019117 * A279918 A273986 A246729

Adjacent sequences: A204369 A204370 A204371 * A204373 A204374 A204375

Keyword

sign

Author

Michael Somos, Jan 14 2012

Status

approved