A160399 - OEIS

%I #42 Sep 08 2022 08:45:45

%S 1,4,11,27,62,137,296,630,1326,2768,5744,11867,24429,50135,102627,

%T 209641,427518,870579,1770536,3596614,7298397,14796658,29974913,

%U 60681233,122767148,248232863,501648844,1013257334,2045684971

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

%C Binomial transform of the sequence d(n) (A000005). - _Emeric Deutsch_, May 15 2009

%C Apparently the partial sums of A101509. - _R. J. Mathar_, May 17 2009

%H Alois P. Heinz, <a href="/A160399/b160399.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric M. Schmidt, <a href="https://dx.doi.org/10.1216/RMJ-2017-47-8-2777">The probability that the number of points on a complete intersection is squarefree</a>, Rocky Mountain J. Math, Volume 47, Number 8 (2017), 2777-2796.

%F G.f.: (Sum_{k>=1} (x/(1-x))^k/(1-x^k/(1-x)^k))/(1-x). - _Emeric Deutsch_, May 15 2009

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

%F 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

%F a(n) = Sum_{i=1..n} Sum_{j=1..n} binomial(n,i*j). - _Ridouane Oudra_, Nov 12 2019

%p with(numtheory): seq(sum(binomial(n, k)*tau(k), k = 1 .. n), n = 1 .. 30); # _Emeric Deutsch_, May 15 2009

%p 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

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

%o (PARI) a(n) = sum(k=1, n, binomial(n, k)*numdiv(k)); \\ _Michel Marcus_, Feb 25 2017

%o (GAP) List([1..10^3], n -> Sum([1..n], k -> Binomial(n,k) * Number(DivisorsInt(k)))); # _Muniru A Asiru_, Feb 04 2018

%o (Magma) [&+[Binomial(n,k)*NumberOfDivisors(k):k in [1..n]]:n in [1..30]]; // _Marius A. Burtea_, Nov 12 2019

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

%Y Cf. A000005. - _Emeric Deutsch_, May 15 2009

%Y Cf. A101509, A185003.

%K nonn

%O 1,2

%A _Leroy Quet_, May 12 2009

%E More terms from _Emeric Deutsch_ and _R. J. Mathar_, May 15 2009