A336976 Expansion of Product_{k>=1} 1/(1 - x^k * (1 + k*x)).

1, 1, 3, 7, 15, 32, 65, 131, 260, 501, 965, 1825, 3419, 6326, 11652, 21230, 38405, 69015, 123334, 218980, 386809, 679757, 1189360, 2071761, 3594325, 6211826, 10698409, 18363038, 31420994, 53605525, 91198970, 154746133, 261929303, 442310873, 745264674, 1253081340, 2102754561

Offset: 0

Author

Seiichi Manyama, Aug 09 2020

Keywords

• nonn

Links

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

Crossrefs

Cf. A227681, A336975, A336977, A336978, A336979, A336980.

Programs

Mathematica

m = 36; CoefficientList[Series[Product[1/(1 - x^k*(1 + k*x)), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 01 2021 *)

PARI

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

PARI

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

Formula

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

a(n) ~ c * phi^(n+1) / sqrt(5), where c = Product_{k>=2} 1/(1 - 1/phi^k*(1 + k/phi)) = 167.5661037860673786430316975350024960626825333609486463342... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 06 2021