A163191 - OEIS

%I #7 Aug 15 2023 02:08:54

%S 1,0,1,8,82,1024,15690,279936,5771363,134218240,3487832978,

%T 100000000000,3138673052884,106993205379072,3937454749863386,

%U 155568096631586816,6568441588686506948,295147905179352825856,14063102470280932000763,708235345355337676357632

%N a(n) = Sum_{k=0..n} (-1)^(n-k)*C(n,k)*sigma(n,k) for n>0 with a(0)=1.

%C Definition: sigma(n,k)= sigma_k(n) = Sum_{d|n} d^k.

%F a(n) = Sum_{d|n} (d-1)^n for n>0 with a(0)=1.

%t a[0] = 1; a[n_] := DivisorSum[n, (#-1)^n &]; Array[a, 20, 0] (* _Amiram Eldar_, Aug 15 2023 *)

%o (PARI) {a(n)=if(n==0,1,sumdiv(n,d,(d-1)^n))}

%o (PARI) {a(n)=if(n==0,1,sum(k=0,n,(-1)^(n-k)*binomial(n,k)*sigma(n,k)))}

%Y Cf. A163190 (variant), A000203 (sigma).

%K nonn

%O 0,4

%A _Paul D. Hanna_, Jul 22 2009