A243938 Expansion of f(-x^5)^10 / f(-x)^2 in powers of x where f() is a Ramanujan theta function.

1, 2, 5, 10, 20, 26, 45, 60, 85, 100, 156, 172, 240, 270, 365, 376, 517, 520, 705, 740, 932, 942, 1260, 1200, 1510, 1580, 1928, 1880, 2420, 2300, 2861, 2864, 3410, 3310, 4265, 3876, 4780, 4740, 5625, 5300, 6672, 6082, 7460, 7270, 8400, 8026, 10092, 9100

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

Euler transform of period 5 sequence [2, 2, 2, 2, -8, ...].

Given g.f. A(x), then B(q) = q^2 * A(q) satisfies 0 = f(B(q), B(q^2), B(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)) = 1/5 (t/i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A243939.

Convolution square of A053723.

Example

G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 26*x^5 + 45*x^6 + 60*x^7 + ...
G.f. = q^2 + 2*q^3 + 5*q^4 + 10*q^5 + 20*q^6 + 26*q^7 + 45*q^8 + 60*q^9 + ...

Mathematica

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

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^5 + A)^10 / eta(x + A)^2, n))};
(Magma) Basis( ModularForms( Gamma0(5), 4), 49) [3];
(Sage) A = ModularForms( Gamma0(5), 4, prec=49) . basis(); (-A[0] + A[2]) / 13;

Crossrefs

Cf. A053723, A243939.

Sequence in context: A285571 A000099 A039690 * A365630 A126105 A117486

Adjacent sequences: A243935 A243936 A243937 * A243939 A243940 A243941

Keyword

nonn

Author

Michael Somos, Jun 15 2014

Status

approved