A356390 - OEIS

%I #18 Aug 18 2022 05:58:52

%S 1,3,17,74,514,3564,30708,250704,2780496,29982240,373350240,

%T 4639870080,67024333440,988156834560,16914631507200,271941778483200,

%U 4999620452198400,94617104704819200,1925772463506124800,39245319872575488000,902004581585737728000

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

%H Seiichi Manyama, <a href="/A356390/b356390.txt">Table of n, a(n) for n = 1..448</a>

%F a(n) = n! * Sum_{k=1..n} A000593(k)/k.

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

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

%F a(n) ~ n! * n * Pi^2/12. - _Vaclav Kotesovec_, Aug 07 2022

%t Table[n! * Sum[Sum[(-1)^(k/d + 1)*d, {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)/k);

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

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

%Y Cf. A356389, A356391.

%Y Cf. A000593, A356010, A356393.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Aug 05 2022