A163190 a(n) = Sum_{k=0..n} C(n,k)*sigma(n,k) for n>0 with a(0)=1.

1, 2, 13, 72, 722, 7808, 122538, 2097280, 43444163, 1000262656, 25997950850, 743008372736, 23312187863060, 793714773262336, 29197324076701082, 1152921975865606144, 48663045048486723204, 2185911559738696663040

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. A000203 (sigma), A007318 (binomial).

Programs

Mathematica

Flatten[{1, Table[Sum[Binomial[n, k] * DivisorSigma[k, n], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Oct 08 2016 *)
a[n_] := DivisorSum[n, (1+#)^n &]; a[0] = 1; Array[a, 18, 0] (* Amiram Eldar, Aug 29 2023 *)

PARI

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

PARI

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

Python

from sympy import divisors
def A163190(n): return sum((1+d)**n for d in divisors(n, generator=True)) if n else 1 # Chai Wah Wu, Nov 21 2023

Formula

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

a(n) ~ exp(1) * n^n. - Vaclav Kotesovec, Oct 08 2016