A254593 a(n) = (3/n^3) * Sum_{d|n} (-1)^(n+d)*moebius(n/d)*binomial(2*d,d).

6, 3, 2, 3, 6, 13, 30, 75, 200, 555, 1590, 4693, 14202, 43863, 137882, 440235, 1424958, 4668304, 15459366, 51692379, 174362770, 592815459, 2030105382, 6998177293, 24270836436, 84646997613, 296744311172, 1045283877639, 3698462401026, 13140509079977, 46869358523238, 167781751129899

Offset: 1

Author

Max Alekseyev, Feb 01 2015

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1650
• R. R. Aidagulov, M. A. Alekseyev. On p-adic approximation of sums of binomial coefficients. Journal of Mathematical Sciences 233:5 (2018), 626-634. doi:10.1007/s10958-018-3948-0 arXiv:1602.02632
• M. Kontsevich, R. Stanley, O. Gorodetsky, et al. A congruence involving binomial coefficients, Mathoverflow, 2015.

Programs

Mathematica

a[n_] := 3/n^3 DivisorSum[n, (-1)^(n+#) MoebiusMu[n/#] Binomial[2#, #]&]; Array[a, 40] (* Jean-François Alcover, Dec 18 2015 *)

PARI

{ a(n) = sumdiv(n,d,(-1)^(n+d)*moebius(n/d)*binomial(2*d,d))*3/n^3 }

Formula

a(n) = 6*A131868(n)/n.

For n == 0, 1, or 3 (mod 4), a(n) = A268592(n)/2; for n == 2 (mod 4), a(n) = A268592(n)/2 + A268592(n/2)/8.