A307676 Expansion of Product_{k>=1} (1 - x^k*(1 - x))/(1 - x^k*(1 + x)).

1, 0, 2, 4, 6, 14, 22, 46, 74, 138, 236, 406, 698, 1182, 1994, 3342, 5590, 9274, 15386, 25380, 41818, 68670, 112586, 184210, 300940, 490962, 800026, 1302278, 2118008, 3442042, 5590092, 9073632, 14720738, 23872776, 38700910, 62720726, 101622398, 164617032

Offset: 0

Author

Seiichi Manyama, Apr 21 2019

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A227681, A198197.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 - x^k*(1 - x))/(1 - x^k*(1 + x)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 31 2021 *)

PARI

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

PARI

N=66; x='x+O('x^N); Vec(exp(sum(k=1, N, x^k*sumdiv(k, d, ((1+x)^d-(1-x)^d)/d))))

Formula

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

a(n) ~ phi^(n+4) / sqrt(5), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jul 31 2021