A352013 - OEIS

%I #26 Nov 14 2024 02:11:03

%S 1,0,3,-11,25,-59,721,-10919,60481,-15119,3628801,-93471839,479001601,

%T -8648639,134399865601,-2833553923199,20922789888001,-174888473759999,

%U 6402373705728001,-228084898487846399,3652732042831872001,-14079294028799

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

%H Robert Israel, <a href="/A352013/b352013.txt">Table of n, a(n) for n = 1..449</a>

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

%F E.g.f.: -Sum_{k>0} (-1)^k * (exp(x^k) - 1)/k. - _Seiichi Manyama_, Jun 18 2023

%p restart;

%p f:= proc(n) local d;

%p   add((-1)^(n/d + 1) * (n-1)!/(d-1)!, d = numtheory:-divisors(n))

%p end proc:

%p map(f, [$1..30]); # _Robert Israel_, Nov 14 2024

%t a[n_] := DivisorSum[n, (-1)^(n/#+1) * (n-1)!/(#-1)! &]; Array[a, 22] (* _Amiram Eldar_, Aug 30 2023 *)

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

%o (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, log(1+x^k)/k!)))

%Y Cf. A087906, A298906, A327243, A352014.

%K sign

%O 1,3

%A _Seiichi Manyama_, Feb 28 2022