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a(n) = Sum_{k=0..n} C(n,k)*sigma(n,k) for n>0 with a(0)=1.
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%I #13 Nov 21 2023 12:18:18

%S 1,2,13,72,722,7808,122538,2097280,43444163,1000262656,25997950850,

%T 743008372736,23312187863060,793714773262336,29197324076701082,

%U 1152921975865606144,48663045048486723204,2185911559738696663040

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

%F a(n) ~ exp(1) * n^n. - _Vaclav Kotesovec_, Oct 08 2016

%t Flatten[{1, Table[Sum[Binomial[n, k] * DivisorSigma[k, n], {k, 0, n}], {n, 1, 20}]}] (* _Vaclav Kotesovec_, Oct 08 2016 *)

%t a[n_] := DivisorSum[n, (1+#)^n &]; a[0] = 1; Array[a, 18, 0] (* _Amiram Eldar_, Aug 29 2023 *)

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

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

%o (Python)

%o from sympy import divisors

%o 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

%Y Cf. A000203 (sigma), A007318 (binomial).

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 22 2009