A029838 -id:A029838 - cp's OEIS frontend

A307484 Expansion of Product_{k>=1} 1/(1+x^k)^((-1)^k*k^2).

1, 1, -4, 5, 3, -17, 33, -61, 67, 63, -392, 803, -1070, 898, 482, -4449, 11362, -18630, 21105, -11067, -24871, 103562, -227004, 359040, -417697, 266106, 312987, -1578543, 3635615, -6157911, 8155892, -7689028, 1502546, 14707881, -44539735, 87849728, -136927058, 171008704

Offset: 0

Seiichi Manyama, Apr 10 2019

sign

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = (-1)^n * n^2, g(n) = -1.

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

Product_{k>=1} 1/(1+x^k)^((-1)^k*k^b): A029838 (b=0), A284467 (b=1), this sequence (b=2).

Cf. A177155, A307462.

m = 37; CoefficientList[Series[Product[1/(1+x^k)^((-1)^k*k^2), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 14 2021 *)
nmax = 40; CoefficientList[Series[Product[(1 + x^(2*k - 1))^((2*k - 1)^2)/(1 + x^(2*k))^(4*k^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 14 2021 *)

N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1+x^k)^((-1)^k*k^2)))

A306575 Expansion of 1/(1 - x - x^2/(1 - x^2 - x^3/(1 - x^3 - x^4/(1 - x^4 - x^5/(1 - ...))))), a continued fraction.

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 21, 40, 77, 148, 285, 550, 1061, 2049, 3957, 7644, 14768, 28535, 55138, 106549, 205902, 397906, 768967, 1486070, 2871932, 5550233, 10726300, 20729542, 40061784, 77423250, 149628008, 289170949, 558851751, 1080037175, 2087280839, 4033881485

Offset: 0

Ilya Gutkovskiy, Jun 25 2019

nonn

Cf. A029838, A083365, A088352.

nmax = 35; CoefficientList[Series[1/(1 - x + ContinuedFractionK[-x^k, 1 - x^k, {k, 2, nmax}]), {x, 0, nmax}], x]

a(n) ~ c * d^n, where d = 1.9326019136649450138850556203... and c = 0.389707331111778150048054243... - Vaclav Kotesovec, Jul 01 2019

cp's OEIS Frontend

A307484 Expansion of Product_{k>=1} 1/(1+x^k)^((-1)^k*k^2).

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