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A205815 a(n) = Sum_{k=1..n} binomial(n,k) * sigma(n,k) * 2^(n-k).

%I A205815 #12 Oct 25 2024 04:49:57
%S A205815 1,17,136,1585,16986,282338,4784900,101750689,2359918963,62200943002,
%T A205815 1792160567088,56765070059074,1946195069937314,72080471103535786,
%U A205815 2862427829603121696,121449533922041845569,5480386857784931063958,262149577935595804303451
%N A205815 a(n) = Sum_{k=1..n} binomial(n,k) * sigma(n,k) * 2^(n-k).
%C A205815 Here sigma(n,k) equals the sum of the k-th powers of the divisors of n.
%F A205815 Logarithmic derivative of A205814.
%F A205815 a(n) = Sum_{d|n} ((d+2)^n - 2^n).
%F A205815 a(n) ~ exp(2) * n^n. - _Vaclav Kotesovec_, Oct 08 2016
%e A205815 L.g.f.: L(x) = x + 17*x^2/2 + 136*x^3/3 + 1585*x^4/4 + 16986*x^5/5 +...
%e A205815 Exponentiation yields the g.f. of A205814:
%e A205815 exp(L(x)) = 1 + x + 9*x^2 + 54*x^3 + 482*x^4 + 4239*x^5 + 55561*x^6 +...
%e A205815 Illustration of terms.
%e A205815 a(2) = 2*sigma(2,1)*2 + 1*sigma(2,2)*1 = 2*3*2 + 1*5*1 = 17;
%e A205815 a(3) = 3*sigma(3,1)*4 + 3*sigma(3,2)*2 + 1*sigma(3,3)*1 = 3*4*4 + 3*10*2 + 1*28*1 = 136;
%e A205815 a(4) = 4*sigma(4,1)*8 + 6*sigma(4,2)*4 + 4*sigma(4,3)*2 + 1*sigma(4,3)*1 = 4*7*8 + 6*21*4 + 4*73*2 + 1*273*1 = 1585.
%t A205815 Table[Sum[Binomial[n, k] * DivisorSigma[k, n] * 2^(n-k), {k, 1, n}], {n, 1, 20}] (* _Vaclav Kotesovec_, Oct 08 2016 *)
%o A205815 (PARI) {a(n)=sum(k=1, n, binomial(n, k)*sigma(n, k)*2^(n-k))}
%Y A205815 Cf. A205814 (exp), A205812.
%K A205815 nonn,easy
%O A205815 1,2
%A A205815 _Paul D. Hanna_, Feb 01 2012