A258724 Expansion of f(-x)^11 / f(-x^3) + 27 * x * f(-x^3)^11 / f(-x) in powers of x where f() is a Ramanujan theta function.

1, 16, 71, 0, -337, 256, -601, 0, 625, 1136, 194, 0, -529, 0, -3214, 0, 2640, -5392, 0, 0, 7199, 4096, 2903, 0, -1249, -9616, 4679, 0, 0, 0, -23927, 0, 9071, 10000, -19849, 0, 22034, 18176, 0, 0, 14641, 3104, -10942, 0, -42671, 0, 24359, 0, 0, -8464, -42121

Offset

0,2

Comments

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

This is a member of an infinite family of integer weight modular forms. g_1 = A033687, g_2 = A030206, g_3 = A130539, g_4 = A000731.

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^(-1/3) * (eta(q)^11 / eta(q^3) + 27 * eta(q^3)^11 / eta(q)) in powers of q.

a(n) = b(3*n + 1) where b(n) is multiplicative and b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^(2*e) if p == 2 (mod 3), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) if p == 1 (mod 3) where b(p) = (y^2 - 2*p)^2 - 2*p^2, 4*p = y^2 + 27*x^2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 27^(5/2) (t/i)^5 f(t) where q = exp(2 Pi i t).

a(4*n + 3) = 0.

Example

G.f. = 1 + 16*x + 71*x^2 - 337*x^4 + 256*x^5 - 601*x^6 + 625*x^8 + ...
G.f. = q + 16*q^4 + 71*q^7 - 337*q^13 + 256*q^16 - 601*q^19 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^11 / QPochhammer[ x^3] + 27 x QPochhammer[ x^3]^11 / QPochhammer[ x], {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^11 / eta(x^3 + A) + 27 * x * eta(x^3 + A)^11 / eta(x + A), n))};
(PARI) {a(n) = my(A, p, e, x, y, a0, a1); n = 3*n + 1; if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 0, p%3==2, if( e%2, 0, p^(2*e)), for( x=1, sqrtint(4*p\27), if( issquare(4*p - 27*x^2, &y), break)); y = (y^2 - 2*p)^2 - 2*p^2; a0=1; a1=y; for( i=2, e, x = y*a1 - p^4*a0; a0=a1; a1=x); a1)))};

Crossrefs

Cf. A000731, A030206, A033687, A130539.

Sequence in context: A220212 A027997 A284844 * A264888 A272964 A232579

Adjacent sequences: A258721 A258722 A258723 * A258725 A258726 A258727

Keyword

sign

Author

Michael Somos, Jun 08 2015

Status

approved