cp's OEIS Frontend

A222115 a(n) = 1 + Sum_{k=1..n} binomial(n,k) * sigma(k).

%I A222115 #65 Dec 30 2015 07:04:28
%S A222115 2,6,17,46,117,285,674,1558,3536,7911,17503,38377,83501,180480,387882,
%T A222115 829606,1766999,3749766,7931115,16724871,35173778,73794661,154485528,
%U A222115 322771345,673155142,1401536935,2913490376,6047714600,12536770559,25956242580,53678385267,110889844998
%N A222115 a(n) = 1 + Sum_{k=1..n} binomial(n,k) * sigma(k).
%C A222115 Here sigma(n) is the sum of divisors of n (A000203).
%H A222115 Paul D. Hanna, <a href="/A222115/b222115.txt">Table of n, a(n) for n = 1..1000</a>
%F A222115 Logarithmic derivative of the binomial transform of the partition numbers (A218481).
%F A222115 L.g.f.: -log(1-x) + Sum_{n>=1} sigma(n) * x^n/(1-x)^n / n.
%F A222115 L.g.f.: -log(1-x) + Sum_{n>=1} x^n/((1-x)^n - x^n) / n.
%F A222115 L.g.f.: -log(1-x) + Sum_{n>=1} n*log(1-x) - log((1-x)^n - x^n).
%F A222115 L.g.f.: -log(1-x) + 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 A222115 a(n) = A185003(n) + 1.
%F A222115 a(n) ~ Pi^2/12 * n * 2^n. - _Vaclav Kotesovec_, Dec 30 2015
%e A222115 L.g.f.: L(x) = 2*x + 6*x^2/2 + 17*x^3/3 + 46*x^4/4 + 117*x^5/5 + 285*x^6/6 +...
%e A222115 where
%e A222115 exp(L(x)) = 1 + 2*x + 5*x^2 + 13*x^3 + 34*x^4 + 88*x^5 + 225*x^6 + 569*x^7 +...+ A218481(n)*x^n +...
%t A222115 Table[Sum[Binomial[n,k]DivisorSigma[1,k],{k,n}],{n,40}]+1 (* _Harvey P. Dale_, Jul 21 2015 *)
%o A222115 (PARI) {a(n)=1+sum(k=1,n,binomial(n,k)*sigma(k))}
%o A222115 for(n=1,30,print1(a(n),", "))
%o A222115 (PARI) {a(n)=local(X=x+x*O(x^n)); n*polcoeff(-log(1-X)+sum(m=1, n+1, x^m/((1-x)^m-X^m)/m), n)}
%o A222115 (PARI) {a(n)=local(X=x+x*O(x^n)); n*polcoeff(-log(1-X)+sum(k=1, n, k*log(1-X)-log((1-x)^k-X^k)), n)}
%o A222115 (PARI) {a(n)=local(X=x+x*O(x^n)); n*polcoeff(-log(1-X)+sum(m=1, n+1, sigma(m)*x^m/(1-X)^m/m), n)}
%o A222115 (PARI) {a(n)=local(X=x+x*O(x^n)); n*polcoeff(-log(1-X)+sum(k=1, n, valuation(2*k, 2)*log(1 + x^k/(1-X)^k)), n)}
%Y A222115 Cf. A218481, A185003, A000203.
%K A222115 nonn
%O A222115 1,1
%A A222115 _Paul D. Hanna_, Jun 01 2013