A268592 a(n) = (6/n^3) * Sum_{d|n} moebius(n/d)*binomial(2*d,d).

12, 3, 4, 6, 12, 25, 60, 150, 400, 1107, 3180, 9386, 28404, 87711, 275764, 880470, 2849916, 9336508, 30918732, 103384758, 348725540, 1185630123, 4060210764, 13996354586, 48541672872, 169293988125, 593488622344, 2090567755278, 7396924802052, 26281018091013, 93738717046476, 335563502259798

Offset: 1

Author

Max Alekseyev, Feb 07 2016

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1500
• 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

Crossrefs

Cf. A254593, A268618, A268619.

Programs

Mathematica

a[n_] := (6/n^3)* DivisorSum[n, MoebiusMu[n/#] Binomial[2 #, #] &]; Array[a, 50] (* G. C. Greubel, Dec 15 2017 *)

PARI

{ a(n) = sumdiv(n, d, moebius(n/d)*binomial(2*d, d))*6/n^3; }

Formula

a(n) = A007727(n)*6/n^3 = A045630(n)*12/n^3 = A060165(n)*6/n^2 = A022553(n)*12/n^2 = A268619(n)*6/n.

For n == 0, 1, or 3 (mod 4), a(n) = 2*A254593(n); for n == 2 (mod 4), a(n) = 2*A254593(n) - A254593(n/2)/2.