A060170 -id:A060170 - cp's OEIS frontend

A268618 a(n) = (2/n^3) * Sum_{d|n} moebius(n/d)binomial(3d,d).

6, 3, 6, 15, 48, 171, 678, 2871, 12858, 60084, 290814, 1448679, 7394106, 38527779, 204365880, 1101000087, 6013054788, 33239486925, 185736687366, 1047961118940, 5964676687668, 34219227608607, 197737647050742, 1150211467134927, 6731334034067058, 39614408616493581, 234342269725331130, 1392933275876114127

Offset: 1

Max Alekseyev, Feb 09 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1200
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

Cf. A268592, A254593

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

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

a(n) = (2/n^3)* Sum_{d|n} A008683(n/d)*A005809(d) = (2/n^2)*A060170(n) = (2/n)*A268617(n).

A268617 a(n) = (1/n^2) * Sum_{d|n} moebius(n/d)binomial(3d,d).

Original entry on oeis.org

3, 3, 9, 30, 120, 513, 2373, 11484, 57861, 300420, 1599477, 8692074, 48061689, 269694453, 1532744100, 8808000696, 51110965698, 299155382325, 1764498529977, 10479611189400, 62629105220514, 376411503694677, 2273982941083533, 13802537605619124, 84141675425838225, 514987312014416553, 3163620641291970255

Offset: 1

Max Alekseyev, Feb 09 2016

nonn

2*a(n) is divisible by n (cf. A268618).

Cf. A005809, A008683, A060170, A268618, A268619.

a[n_] := DivisorSum[n, MoebiusMu[n/#] * Binomial[3*#, #] &] / n^2; Array[a, 30] (* Amiram Eldar, Aug 24 2023 *)

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

a(n) = (1/n^2)* Sum_{d|n} A008683(n/d)*A005809(d).

a(n) = A060170(n) / n = A268618(n)*n/2.

cp's OEIS Frontend

A268618 a(n) = (2/n^3) * Sum_{d|n} moebius(n/d)binomial(3d,d).

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A268617 a(n) = (1/n^2) * Sum_{d|n} moebius(n/d)binomial(3d,d).

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cp's OEIS Frontend

A268618 a(n) = (2/n^3) * Sum_{d|n} moebius(n/d)*binomial(3*d,d).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A268617 a(n) = (1/n^2) * Sum_{d|n} moebius(n/d)*binomial(3*d,d).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

A268618 a(n) = (2/n^3) * Sum_{d|n} moebius(n/d)binomial(3d,d).

A268617 a(n) = (1/n^2) * Sum_{d|n} moebius(n/d)binomial(3d,d).