A205815 a(n) = Sum_{k=1..n} binomial(n,k) * sigma(n,k) * 2^(n-k).

1, 17, 136, 1585, 16986, 282338, 4784900, 101750689, 2359918963, 62200943002, 1792160567088, 56765070059074, 1946195069937314, 72080471103535786, 2862427829603121696, 121449533922041845569, 5480386857784931063958, 262149577935595804303451

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..18.

Formula

Logarithmic derivative of A205814.

a(n) = Sum_{d|n} ((d+2)^n - 2^n).

a(n) ~ exp(2) * n^n. - Vaclav Kotesovec, Oct 08 2016

Example

L.g.f.: L(x) = x + 17*x^2/2 + 136*x^3/3 + 1585*x^4/4 + 16986*x^5/5 +...
Exponentiation yields the g.f. of A205814:
exp(L(x)) = 1 + x + 9*x^2 + 54*x^3 + 482*x^4 + 4239*x^5 + 55561*x^6 +...
Illustration of terms.
a(2) = 2*sigma(2,1)*2 + 1*sigma(2,2)*1 = 2*3*2 + 1*5*1 = 17;
a(3) = 3*sigma(3,1)*4 + 3*sigma(3,2)*2 + 1*sigma(3,3)*1 = 3*4*4 + 3*10*2 + 1*28*1 = 136;
a(4) = 4*sigma(4,1)*8 + 6*sigma(4,2)*4 + 4*sigma(4,3)*2 + 1*sigma(4,3)*1 = 4*7*8 + 6*21*4 + 4*73*2 + 1*273*1 = 1585.

Mathematica

Table[Sum[Binomial[n, k] * DivisorSigma[k, n] * 2^(n-k), {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Oct 08 2016 *)

Prog

(PARI) {a(n)=sum(k=1, n, binomial(n, k)*sigma(n, k)*2^(n-k))}

Crossrefs

Cf. A205814 (exp), A205812.

Sequence in context: A047642 A010933 A022612 * A060220 A041550 A142788

Adjacent sequences: A205812 A205813 A205814 * A205816 A205817 A205818

Keyword

nonn,easy

Author

Paul D. Hanna, Feb 01 2012

Status

approved