A160399 a(n) = Sum_{k=1..n} binomial(n,k) * d(k), where d(k) = the number of positive divisors of k.

1, 4, 11, 27, 62, 137, 296, 630, 1326, 2768, 5744, 11867, 24429, 50135, 102627, 209641, 427518, 870579, 1770536, 3596614, 7298397, 14796658, 29974913, 60681233, 122767148, 248232863, 501648844, 1013257334, 2045684971

Offset: 1

Author

Leroy Quet, May 12 2009

Keywords

• nonn

Comments

Binomial transform of the sequence d(n) (A000005). - Emeric Deutsch, May 15 2009

Apparently the partial sums of A101509. - R. J. Mathar, May 17 2009

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Eric M. Schmidt, The probability that the number of points on a complete intersection is squarefree, Rocky Mountain J. Math, Volume 47, Number 8 (2017), 2777-2796.

Crossrefs

Cf. A000005. - Emeric Deutsch, May 15 2009

Cf. A101509, A185003.

Programs

GAP

List([1..10^3], n -> Sum([1..n], k -> Binomial(n,k) * Number(DivisorsInt(k)))); # Muniru A Asiru, Feb 04 2018

Magma

[&+[Binomial(n,k)*NumberOfDivisors(k):k in [1..n]]:n in [1..30]]; // Marius A. Burtea, Nov 12 2019

Magma

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

Maple

with(numtheory): seq(sum(binomial(n, k)*tau(k), k = 1 .. n), n = 1 .. 30); # Emeric Deutsch, May 15 2009
A160399 := proc(n) local k; add(binomial(n,k)*numtheory[tau](k),k=1..n) ; end: seq(A160399(n),n=1..40) ; # R. J. Mathar, May 17 2009

Mathematica

a[n_] := Sum[Binomial[n, k]*DivisorSigma[0, k], {k, 1, n}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Feb 25 2017 *)

PARI

a(n) = sum(k=1, n, binomial(n, k)*numdiv(k)); \\ Michel Marcus, Feb 25 2017

Formula

G.f.: (Sum_{k>=1} (x/(1-x))^k/(1-x^k/(1-x)^k))/(1-x). - Emeric Deutsch, May 15 2009

E.g.f.: exp(x)*Sum_{k>=1} d(k)*x^k/k!. - Ilya Gutkovskiy, Nov 26 2017

a(n) = 2^n*(log(n) + 2*gamma - log(2)) + O(2^n*n^(-1/4)). [Put alpha_n = beta_n = 1/2 in Thm. 4.2 of Schmidt.] - Eric M. Schmidt, Feb 03 2018

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

Extensions

More terms from Emeric Deutsch and R. J. Mathar, May 15 2009