OFFSET
1,1
COMMENTS
Here sigma(n) is the sum of divisors of n (A000203).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..1000
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
EXAMPLE
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 +...
MATHEMATICA
Table[Sum[Binomial[n, k]DivisorSigma[1, k], {k, n}], {n, 40}]+1 (* Harvey P. Dale, Jul 21 2015 *)
PROG
(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)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 01 2013
STATUS
approved