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A205812 a(n) = Sum_{k=1..n} binomial(n,k) * sigma(n,k).

%I A205812 #21 Oct 25 2024 06:25:57
%S A205812 1,11,70,719,7806,122534,2097278,43444159,1000262653,25997950846,
%T A205812 743008372734,23312187863054,793714773262334,29197324076701078,
%U A205812 1152921975865606140,48663045048486723199,2185911559738696663038,104128351926393946602653,5242880000000000000524286
%N A205812 a(n) = Sum_{k=1..n} binomial(n,k) * sigma(n,k).
%C A205812 Here sigma(n,k) equals the sum of the k-th powers of the divisors of n.
%F A205812 Logarithmic derivative of A205811.
%F A205812 a(n) = Sum_{d|n} ((d+1)^n - 1).
%F A205812 a(n) = A163190(n) - tau(n).
%F A205812 a(n) ~ exp(1) * n^n. - _Vaclav Kotesovec_, Oct 08 2016
%e A205812 L.g.f.: L(x) = x + 11*x^2/2 + 70*x^3/3 + 719*x^4/4 + 7806*x^5/5 +...
%e A205812 Exponentiation yields the g.f. of A205811:
%e A205812 exp(L(x)) = 1 + x + 6*x^2 + 29*x^3 + 221*x^4 + 1897*x^5 + 23502*x^6 +...
%e A205812 Illustration of terms.
%e A205812 a(2) = 2*sigma(2,1) + 1*sigma(2,2) = 2*3 + 1*5 = 11;
%e A205812 a(3) = 3*sigma(3,1) + 3*sigma(3,2) + 1*sigma(3,3) = 3*4 + 3*10 + 1*28 = 70;
%e A205812 a(4) = 4*sigma(4,1) + 6*sigma(4,2) + 4*sigma(4,3) + 1*sigma(4,3) = 4*7 + 6*21 + 4*73 + 1*273 = 719.
%t A205812 Table[Sum[Binomial[n, k]*DivisorSigma[k, n], {k, 1, n}], {n, 1, 20}] (* _Vaclav Kotesovec_, Oct 08 2016 *)
%o A205812 (PARI) {a(n)=sum(k=1,n,binomial(n,k)*sigma(n,k))}
%Y A205812 Cf. A000203, A205811, A163190, A205815, A377331.
%K A205812 nonn
%O A205812 1,2
%A A205812 _Paul D. Hanna_, Feb 01 2012