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

%I A163190 #13 Nov 21 2023 12:18:18
%S A163190 1,2,13,72,722,7808,122538,2097280,43444163,1000262656,25997950850,
%T A163190 743008372736,23312187863060,793714773262336,29197324076701082,
%U A163190 1152921975865606144,48663045048486723204,2185911559738696663040
%N A163190 a(n) = Sum_{k=0..n} C(n,k)*sigma(n,k) for n>0 with a(0)=1.
%C A163190 Definition: sigma(n,k) = sigma_k(n) = Sum_{d|n} d^k.
%F A163190 a(n) = Sum_{d|n} (1+d)^n for n>0 with a(0)=1.
%F A163190 a(n) ~ exp(1) * n^n. - _Vaclav Kotesovec_, Oct 08 2016
%t A163190 Flatten[{1, Table[Sum[Binomial[n, k] * DivisorSigma[k, n], {k, 0, n}], {n, 1, 20}]}] (* _Vaclav Kotesovec_, Oct 08 2016 *)
%t A163190 a[n_] := DivisorSum[n, (1+#)^n &]; a[0] = 1; Array[a, 18, 0] (* _Amiram Eldar_, Aug 29 2023 *)
%o A163190 (PARI) {a(n)=if(n==0,1,sumdiv(n,d,(1+d)^n))}
%o A163190 (PARI) {a(n)=if(n==0,1,sum(k=0,n,binomial(n,k)*sigma(n,k)))}
%o A163190 (Python)
%o A163190 from sympy import divisors
%o A163190 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 A163190 Cf. A000203 (sigma), A007318 (binomial).
%K A163190 nonn
%O A163190 0,2
%A A163190 _Paul D. Hanna_, Jul 22 2009