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A185003 a(n) = Sum_{k=1..n} binomial(n,k)*sigma(k).

%I A185003 #30 Sep 08 2022 08:45:55
%S A185003 1,5,16,45,116,284,673,1557,3535,7910,17502,38376,83500,180479,387881,
%T A185003 829605,1766998,3749765,7931114,16724870,35173777,73794660,154485527,
%U A185003 322771344,673155141,1401536934,2913490375,6047714599,12536770558,25956242579,53678385266
%N A185003 a(n) = Sum_{k=1..n} binomial(n,k)*sigma(k).
%H A185003 G. C. Greubel, <a href="/A185003/b185003.txt">Table of n, a(n) for n = 1..1000</a>
%F A185003 Logarithmic derivative of A103446 (with offset=0), which describes the binomial transform of partitions.
%F A185003 From _Paul D. Hanna_, Jun 01 2013: (Start)
%F A185003 L.g.f.: Sum_{n>=1} sigma(n) * x^n/(1-x)^n / n.
%F A185003 L.g.f.: Sum_{n>=1} x^n/((1-x)^n - x^n) / n.
%F A185003 L.g.f.: Sum_{n>=1} n*log(1-x) - log((1-x)^n - x^n).
%F A185003 L.g.f.: Sum_{n>=1} A001511(n) * log(1 + x^n/(1-x)^n), where 2^A001511(n) is the highest power of 2 that divides 2*n.
%F A185003 a(n) = A222115(n) - 1. (End)
%F A185003 a(n) ~ Pi^2/12 * n * 2^n. - _Vaclav Kotesovec_, Dec 30 2015
%F A185003 a(n) = Sum_{i=1..n} Sum_{j=1..n} i*binomial(n,i*j). - _Ridouane Oudra_, Nov 12 2019
%e A185003 L.g.f.: L(x) = x + 5*x^2/2 + 16*x^3/3 + 45*x^4/4 + 116*x^5/5 + 284*x^6/6 +...
%e A185003 where exponentiation yields A103446 (with offset=0):
%e A185003 exp(L(x)) = 1 + x + 3*x^2 + 8*x^3 + 21*x^4 + 54*x^5 + 137*x^6 + 344*x^7 +...
%p A185003 with(numtheory): seq(add(binomial(n,i)*sigma(i), i=1..n), n=1..40); # _Ridouane Oudra_, Nov 12 2019
%t A185003 Table[Sum[Binomial[n, k] DivisorSigma[1, k], {k, n}], {n,50}] (* _G. C. Greubel_, Jun 03 2017 *)
%o A185003 (PARI) {a(n)=sum(k=1,n,sigma(k)*binomial(n,k))}
%o A185003 for(n=1,30,print1(a(n),", "))
%o A185003 (PARI) {a(n)=local(X=x+x*O(x^n)); n*polcoeff(sum(m=1, n+1, x^m/((1-x)^m-X^m)/m), n)}
%o A185003 (PARI) {a(n)=local(X=x+x*O(x^n)); n*polcoeff(sum(k=1, n, k*log(1-X)-log((1-x)^k-X^k)), n)}
%o A185003 (PARI) {a(n)=local(X=x+x*O(x^n)); n*polcoeff(sum(m=1, n+1, sigma(m)*x^m/(1-X)^m/m), n)}
%o A185003 (PARI) {a(n)=local(X=x+x*O(x^n)); n*polcoeff(sum(k=1, n, valuation(2*k, 2)*log(1 + x^k/(1-X)^k)), n)}
%o A185003 (Magma) [&+[Binomial(n,k)*DivisorSigma(1,k):k in [1..n]]:n in [1..31]]; // _Marius A. Burtea_, Nov 12 2019
%o A185003 (Magma) [&+[&+[i*Binomial(n,i*j):j in [1..n]]:i in [1..n]]:n in [1..31]]; // _Marius A. Burtea_, Nov 12 2019
%Y A185003 Cf. A103446, A222115, A000203, A160399.
%K A185003 nonn
%O A185003 1,2
%A A185003 _Paul D. Hanna_, Feb 04 2012