A205812 a(n) = Sum_{k=1..n} binomial(n,k) * sigma(n,k).

1, 11, 70, 719, 7806, 122534, 2097278, 43444159, 1000262653, 25997950846, 743008372734, 23312187863054, 793714773262334, 29197324076701078, 1152921975865606140, 48663045048486723199, 2185911559738696663038, 104128351926393946602653, 5242880000000000000524286

Offset: 1

Author

Paul D. Hanna, Feb 01 2012

Keywords

• nonn

Comments

Here sigma(n,k) equals the sum of the k-th powers of the divisors of n.

Examples

			L.g.f.: L(x) = x + 11*x^2/2 + 70*x^3/3 + 719*x^4/4 + 7806*x^5/5 +...
Exponentiation yields the g.f. of A205811:
exp(L(x)) = 1 + x + 6*x^2 + 29*x^3 + 221*x^4 + 1897*x^5 + 23502*x^6 +...
Illustration of terms.
a(2) = 2*sigma(2,1) + 1*sigma(2,2) = 2*3 + 1*5 = 11;
a(3) = 3*sigma(3,1) + 3*sigma(3,2) + 1*sigma(3,3) = 3*4 + 3*10 + 1*28 = 70;
a(4) = 4*sigma(4,1) + 6*sigma(4,2) + 4*sigma(4,3) + 1*sigma(4,3) = 4*7 + 6*21 + 4*73 + 1*273 = 719.
		

Crossrefs

Cf. A000203, A205811, A163190, A205815, A377331.

Programs

Mathematica

Table[Sum[Binomial[n, k]*DivisorSigma[k, n], {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Oct 08 2016 *)

PARI

{a(n)=sum(k=1,n,binomial(n,k)*sigma(n,k))}

Formula

Logarithmic derivative of A205811.

a(n) = Sum_{d|n} ((d+1)^n - 1).

a(n) = A163190(n) - tau(n).

a(n) ~ exp(1) * n^n. - Vaclav Kotesovec, Oct 08 2016