A307520 -id:A307520 - cp's OEIS frontend

A307521 Expansion of Product_{k>=1} ((1 + x)^k + x^k)/((1 + x)^k - x^k).

1, 2, 2, 2, 0, 2, -2, 2, 0, -4, 8, -4, -26, 112, -288, 560, -832, 782, 274, -3378, 9424, -17498, 21182, -2154, -78180, 284594, -700018, 1381802, -2250316, 2877674, -2172870, -1955998, 12715122, -33812990, 67322842, -108956110, 139447006, -110023870, -83188990, 651268018

Offset: 0

Seiichi Manyama, Apr 12 2019

sign

Cf. A002448, A307520, A307522, A318570.

m = 39; CoefficientList[Series[Product[((1 + x)^k + x^k)/((1 + x)^k - x^k), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 14 2021 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1+x)^k+x^k)/((1+x)^k-x^k)))

G.f.: 1/theta_4(x/(1 + x)), where theta_4() is the Jacobi theta function.

A307522 Expansion of Product_{k>=1} ((1 + x)^k - x^k)/((1 + x)^k + x^k).

Original entry on oeis.org

1, -2, 2, -2, 4, -10, 22, -42, 72, -116, 188, -332, 662, -1432, 3148, -6736, 13784, -26894, 50254, -90782, 160856, -285230, 518170, -983710, 1964800, -4090002, 8705322, -18582722, 39219572, -81148034, 163946630, -323136562, 622125982, -1173528562, 2179230066

Offset: 0

Seiichi Manyama, Apr 12 2019

Cf. A002448, A103198, A307520, A307521, A318570.

a(n) := 2*(-1)^n*add( binomial(n-1, n-k^2), k = 1..floor(sqrt(n))):
print(1, seq(a(n), n = 1..40)); # Peter Bala, Dec 31 2024

m = 34; CoefficientList[Series[Product[((1 + x)^k - x^k)/((1 + x)^k + x^k), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 14 2021 *)

my(N=66, x='x+O('x^N)); Vec(prod(k=1, N, ((1+x)^k-x^k)/((1+x)^k+x^k)))

G.f.: theta_4(x/(1 + x)), where theta_4() is the Jacobi theta function.
From Peter Bala, Dec 31 2024: (Start)
For n >= 1, a(n) = 2 * (-1)^n * Sum_{k = 1..floor(sqrt(n))} binomial(n-1, n-k^2).
For n >= 1, |a(n)| = 2 * A103198(n). (End)

cp's OEIS Frontend

A307521 Expansion of Product_{k>=1} ((1 + x)^k + x^k)/((1 + x)^k - x^k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula