A356391 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d + 1) * d^2 ) /k.

1, 5, 35, 206, 1654, 13524, 130668, 1262064, 15027696, 178581600, 2407111200, 33276182400, 514020643200, 8130342124800, 144621487584000, 2537556118272000, 49206063078144000, 982811803276800000, 20991083543732736000, 454612169591580672000, 10763306565511514112000

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..448

Formula

a(n) = n! * Sum_{k=1..n} A078306(k)/k.

E.g.f.: -(1/(1-x)) * Sum_{k>0} (-x)^k/(k * (1 - x^k)^2).

E.g.f.: (1/(1-x)) * Sum_{k>0} k * log(1 + x^k).

a(n) ~ n! * n^2 * 3 * zeta(3) / 8. - Vaclav Kotesovec, Aug 07 2022

Mathematica

Table[n! * Sum[Sum[(-1)^(k/d + 1)*d^2, {d, Divisors[k]}]/k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 07 2022 *)

Prog

(PARI) a(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d^2)/k);
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, (-x)^k/(k*(1-x^k)^2))/(1-x)))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, k*log(1+x^k))/(1-x)))

Crossrefs

Cf. A356389, A356390.

Cf. A078306, A356298, A356394.

Sequence in context: A123008 A038143 A352404 * A221578 A320071 A024062

Adjacent sequences: A356388 A356389 A356390 * A356392 A356393 A356394

Keyword

nonn

Author

Seiichi Manyama, Aug 05 2022

Status

approved