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A160399
a(n) = Sum_{k=1..n} binomial(n,k) * d(k), where d(k) = the number of positive divisors of k.
13
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
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
Sequence in context: A305119 A035593 A295056 * A119706 A034345 A036890
KEYWORD
nonn
AUTHOR
Leroy Quet, May 12 2009
EXTENSIONS
More terms from Emeric Deutsch and R. J. Mathar, May 15 2009
STATUS
approved