A356389 - OEIS

%I #19 Aug 18 2022 05:59:06

%S 1,2,10,34,218,1308,10596,74688,793152,7931520,94504320,1054218240,

%T 14662840320,205279764480,3427909632000,50923531008000,

%U 907545606912000,16335820924416000,323185344975360000,6220416698689536000,140360358705186816000,3087927891514109952000

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

%H Seiichi Manyama, <a href="/A356389/b356389.txt">Table of n, a(n) for n = 1..449</a>

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

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

%F 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

%t Table[n! * Sum[Sum[-(-1)^(k/d), {d, Divisors[k]}]/k, {k, 1, n}], {n, 1, 25}] (* _Vaclav Kotesovec_, Aug 07 2022 *)

%t Table[n! * Sum[(2*DivisorSigma[0, 2*k] - 3*DivisorSigma[0, k])/k, {k, 1, n}], {n, 1, 25}] (* _Vaclav Kotesovec_, Aug 07 2022 *)

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

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

%Y Cf. A356390, A356391.

%Y Cf. A048272, A356297, A356392.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Aug 05 2022