A163191 a(n) = Sum_{k=0..n} (-1)^(n-k)*C(n,k)*sigma(n,k) for n>0 with a(0)=1.

1, 0, 1, 8, 82, 1024, 15690, 279936, 5771363, 134218240, 3487832978, 100000000000, 3138673052884, 106993205379072, 3937454749863386, 155568096631586816, 6568441588686506948, 295147905179352825856, 14063102470280932000763, 708235345355337676357632

Offset: 0

Author

Paul D. Hanna, Jul 22 2009

Keywords

• nonn

Comments

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

Crossrefs

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

Programs

Mathematica

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

PARI

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

PARI

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

Formula

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