A222115 a(n) = 1 + Sum_{k=1..n} binomial(n,k) * sigma(k).
2, 6, 17, 46, 117, 285, 674, 1558, 3536, 7911, 17503, 38377, 83501, 180480, 387882, 829606, 1766999, 3749766, 7931115, 16724871, 35173778, 73794661, 154485528, 322771345, 673155142, 1401536935, 2913490376, 6047714600, 12536770559, 25956242580, 53678385267, 110889844998
Offset: 1
Keywords
Examples
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 +... where 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 +...
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..1000
Programs
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Mathematica
Table[Sum[Binomial[n,k]DivisorSigma[1,k],{k,n}],{n,40}]+1 (* Harvey P. Dale, Jul 21 2015 *) -
PARI
{a(n)=1+sum(k=1,n,binomial(n,k)*sigma(k))} for(n=1,30,print1(a(n),", ")) -
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)} -
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)} -
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)} -
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)}
Formula
Logarithmic derivative of the binomial transform of the partition numbers (A218481).
L.g.f.: -log(1-x) + Sum_{n>=1} sigma(n) * x^n/(1-x)^n / n.
L.g.f.: -log(1-x) + Sum_{n>=1} x^n/((1-x)^n - x^n) / n.
L.g.f.: -log(1-x) + Sum_{n>=1} n*log(1-x) - log((1-x)^n - x^n).
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.
a(n) = A185003(n) + 1.
a(n) ~ Pi^2/12 * n * 2^n. - Vaclav Kotesovec, Dec 30 2015
Comments