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A205812 a(n) = Sum_{k=1..n} binomial(n,k) * sigma(n,k). 7
1, 11, 70, 719, 7806, 122534, 2097278, 43444159, 1000262653, 25997950846, 743008372734, 23312187863054, 793714773262334, 29197324076701078, 1152921975865606140, 48663045048486723199, 2185911559738696663038, 104128351926393946602653, 5242880000000000000524286 (list; graph; refs; listen; history; text; internal format)
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..19.

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

EXAMPLE

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.

MATHEMATICA

Table[Sum[Binomial[n, k]*DivisorSigma[k, n], {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))}

CROSSREFS

Cf. A205811, A163190, A205815.

Sequence in context: A295074 A173200 A071746 * A162568 A255205 A117047

Adjacent sequences:  A205809 A205810 A205811 * A205813 A205814 A205815

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 01 2012

STATUS

approved

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Last modified December 5 23:39 EST 2019. Contains 329784 sequences. (Running on oeis4.)