A336975 Expansion of Product_{k>=1} 1/(1 - x^k * (k + x)).

1, 1, 4, 9, 22, 47, 107, 221, 468, 953, 1932, 3814, 7560, 14625, 28192, 53757, 101827, 190907, 356362, 659716, 1215314, 2224968, 4053914, 7346367, 13260001, 23822114, 42629786, 75991017, 134991954, 238948942, 421656911, 741750026, 1301116634, 2275985891, 3971022904

Offset

0,3

Links

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

Formula

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

a(n) ~ c * n * phi^(n+1) / 5, where c = Product_{k>=3} 1/(1 - 1/phi^k*(k + 1/phi)) = 167.5661037860673786430316975350024960626825333609486463342... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 06 2021

Mathematica

m = 34; CoefficientList[Series[Product[1/(1 - x^k*(k + x)), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 01 2021 *)

Prog

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

Crossrefs

Cf. A006906, A227681, A336976, A336977, A336978, A336979, A336980.

Sequence in context: A002835 A253289 A032288 * A076859 A042833 A048654

Adjacent sequences: A336972 A336973 A336974 * A336976 A336977 A336978

Keyword

nonn

Author

Seiichi Manyama, Aug 09 2020

Status

approved