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A185003 a(n) = Sum_{k=1..n} binomial(n,k)*sigma(k). 15
1, 5, 16, 45, 116, 284, 673, 1557, 3535, 7910, 17502, 38376, 83500, 180479, 387881, 829605, 1766998, 3749765, 7931114, 16724870, 35173777, 73794660, 154485527, 322771344, 673155141, 1401536934, 2913490375, 6047714599, 12536770558, 25956242579, 53678385266 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

FORMULA

Logarithmic derivative of A103446 (with offset=0), which describes the binomial transform of partitions.

From Paul D. Hanna, Jun 01 2013: (Start)

L.g.f.: Sum_{n>=1} sigma(n) * x^n/(1-x)^n / n.

L.g.f.: Sum_{n>=1} x^n/((1-x)^n - x^n) / n.

L.g.f.: Sum_{n>=1} n*log(1-x) - log((1-x)^n - x^n).

L.g.f.: 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) = A222115(n) - 1. (End)

a(n) ~ Pi^2/12 * n * 2^n. - Vaclav Kotesovec, Dec 30 2015

a(n) = Sum_{i=1..n} Sum_{j=1..n} i*binomial(n,i*j). - Ridouane Oudra, Nov 12 2019

EXAMPLE

L.g.f.: L(x) = x + 5*x^2/2 + 16*x^3/3 + 45*x^4/4 + 116*x^5/5 + 284*x^6/6 +...

where exponentiation yields A103446 (with offset=0):

exp(L(x)) = 1 + x + 3*x^2 + 8*x^3 + 21*x^4 + 54*x^5 + 137*x^6 + 344*x^7 +...

MAPLE

with(numtheory): seq(add(binomial(n, i)*sigma(i), i=1..n), n=1..40); # Ridouane Oudra, Nov 12 2019

MATHEMATICA

Table[Sum[Binomial[n, k] DivisorSigma[1, k], {k, n}], {n, 50}] (* G. C. Greubel, Jun 03 2017 *)

PROG

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

for(n=1, 30, print1(a(n), ", "))

(PARI) {a(n)=local(X=x+x*O(x^n)); n*polcoeff(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(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(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(sum(k=1, n, valuation(2*k, 2)*log(1 + x^k/(1-X)^k)), n)}

(Magma) [&+[Binomial(n, k)*DivisorSigma(1, k):k in [1..n]]:n in [1..31]]; // Marius A. Burtea, Nov 12 2019

(Magma) [&+[&+[i*Binomial(n, i*j):j in [1..n]]:i in [1..n]]:n in [1..31]]; // Marius A. Burtea, Nov 12 2019

CROSSREFS

Cf. A103446, A222115, A000203, A160399.

Sequence in context: A270134 A269754 A282425 * A189390 A099327 A004146

Adjacent sequences: A185000 A185001 A185002 * A185004 A185005 A185006

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 04 2012

STATUS

approved

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Last modified December 6 10:22 EST 2022. Contains 358630 sequences. (Running on oeis4.)