A320908 Expansion of Product_{k>=1} theta_4(x^k), where theta_4() is the Jacobi theta function.

1, -2, -2, 2, 4, 6, -6, -2, -8, -12, 2, 6, 20, 14, 22, -2, -14, -34, -20, -42, -48, 34, 10, 50, 48, 80, 82, 52, -16, -30, -142, -130, -138, -226, -54, -70, 80, 190, 310, 238, 392, 178, 178, 86, -40, -148, -582, -506, -546, -680, -656, -126, -336, 262, 428, 930

Offset

0,2

Comments

Convolution of A288007 and A288098.

Convolution inverse of A301554.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Formula

G.f.: Product_{i>=1, j>=1} (1 - x^(i*j))/(1 + x^(i*j)).

G.f.: Product_{k>=1} ((1 - x^k)/(1 + x^k))^d(k), where d(k) is the number of divisors of k (A000005).

G.f.: exp(-Sum_{k>=1} sigma(k)*x^k*(2 + x^k)/(k*(1 - x^(2*k)))).

Maple

with(numtheory): seq(coeff(series(mul(((1-x^k)/(1+x^k))^tau(k), k=1..n), x, n+1), x, n), n = 0 .. 60); # Muniru A Asiru, Oct 23 2018

Mathematica

nmax = 55; CoefficientList[Series[Product[EllipticTheta[4, 0, x^k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 55; CoefficientList[Series[Product[((1 - x^k)/(1 + x^k))^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 55; CoefficientList[Series[Exp[-Sum[DivisorSigma[1, k] x^k (2 + x^k)/(k (1 - x^(2 k))), {k, 1, nmax}]], {x, 0, nmax}], x]

Prog

(PARI) N=99; x='x+O('x^N); Vec(prod(k=1, N, ((1-x^k)/(1+x^k))^numdiv(k))) \\ Seiichi Manyama, Oct 25 2018
(Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(&*[(1-x^(j*k))/(1+x^(j*k)):j in [1..2*m]]): k in [1..2*m]]) )); // G. C. Greubel, Oct 29 2018

Crossrefs

Cf. A000005, A000203, A002448, A288007, A288098, A301554, A320067, A320068.

Sequence in context: A086420 A328106 A342336 * A356692 A361404 A248781

Adjacent sequences: A320905 A320906 A320907 * A320909 A320910 A320911

Keyword

sign,look

Author

Ilya Gutkovskiy, Oct 23 2018

Status

approved