A356389 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d + 1) ) /k.

1, 2, 10, 34, 218, 1308, 10596, 74688, 793152, 7931520, 94504320, 1054218240, 14662840320, 205279764480, 3427909632000, 50923531008000, 907545606912000, 16335820924416000, 323185344975360000, 6220416698689536000, 140360358705186816000, 3087927891514109952000

Offset: 1

Author

Seiichi Manyama, Aug 05 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..449

Crossrefs

Cf. A356390, A356391.

Cf. A048272, A356297, A356392.

Programs

Mathematica

Table[n! * Sum[Sum[-(-1)^(k/d), {d, Divisors[k]}]/k, {k, 1, n}], {n, 1, 25}] (* Vaclav Kotesovec, Aug 07 2022 *)
Table[n! * Sum[(2*DivisorSigma[0, 2*k] - 3*DivisorSigma[0, k])/k, {k, 1, n}], {n, 1, 25}] (* Vaclav Kotesovec, Aug 07 2022 *)

PARI

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

PARI

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

Formula

a(n) = n! * Sum_{k=1..n} A048272(k)/k.

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

a(n) ~ n! * log(2) * (log(n) + 2*gamma - log(2)/2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Aug 07 2022