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

1,2

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.

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

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 *)

Prog

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

Crossrefs

Cf. A000005. - Emeric Deutsch, May 15 2009

Cf. A101509, A185003.

Sequence in context: A305119 A035593 A295056 * A119706 A034345 A036890

Adjacent sequences: A160396 A160397 A160398 * A160400 A160401 A160402

Keyword

nonn

Author

Leroy Quet, May 12 2009

Extensions

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

Status

approved