%I #24 Aug 18 2022 05:58:36

%S 1,5,35,206,1654,13524,130668,1262064,15027696,178581600,2407111200,

%T 33276182400,514020643200,8130342124800,144621487584000,

%U 2537556118272000,49206063078144000,982811803276800000,20991083543732736000,454612169591580672000,10763306565511514112000

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

%H Seiichi Manyama, <a href="/A356391/b356391.txt">Table of n, a(n) for n = 1..448</a>

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

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

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

%F a(n) ~ n! * n^2 * 3 * zeta(3) / 8. - _Vaclav Kotesovec_, Aug 07 2022

%t 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 *)

%o (PARI) a(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d^2)/k);

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, (-x)^k/(k*(1-x^k)^2))/(1-x)))

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, k*log(1+x^k))/(1-x)))

%Y Cf. A356389, A356390.

%Y Cf. A078306, A356298, A356394.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Aug 05 2022