A243939 Expansion of f(-q)^10 / f(-q^5)^2 in power of q where f() is a Ramanujan theta function.

1, -10, 35, -30, -105, 240, -20, -190, -225, -70, 1535, -820, -940, -480, -960, 5470, -1185, -2140, -3505, -3600, 14395, -3820, -3380, -3930, -6300, 23990, -9070, -6900, -10120, -8900, 47980, -10420, -16865, -14960, -16010, 66310, -19485, -22040, -19900

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..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of eta(q)^10 / eta(q^5)^2 in powers of q.

Euler transform of period 5 sequence [-10, -10, -10, -10, -8, ...].

Given g.f. A(q), then 0 = f(A(q), A(q^2), A(q^4)) where f(u, v, w) = (v^3 + u^2*w + 16*u*w^2)^2 - 4*u*w * (u + 2*v) * (v + 8*w) * (v^2 + 2*u*w).

G.f. is a period 1 Fourier series which satisfies f(-1 / (5 t)) = 5^5 (t/i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A243938.

Convolution square of A109064.

Example

G.f. = 1 - 10*q + 35*q^2 - 30*q^3 - 105*q^4 + 240*q^5 - 20*q^6 - 190*q^7 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^10 / QPochhammer[ q^5]^2, {q, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^10 / eta(x^5 + A)^2, n))};
(Magma) A := Basis( ModularForms(5, 4), 12); A[1] - 10*A[2] + 35*A[3];
(Sage) A = ModularForms( Gamma0(5), 4, prec=36) . basis(); A[1] - (A[2] + 25*A[0]) * 5/13;

Crossrefs

Cf. A109064, A243938.

Sequence in context: A254674 A260336 A010818 * A065195 A115149 A044087

Adjacent sequences: A243936 A243937 A243938 * A243940 A243941 A243942

Keyword

sign

Author

Michael Somos, Jun 15 2014

Status

approved