A177837 a(n) = binomial(n^3-2, n-2).

1, 25, 1891, 302621, 84957251, 37307689133, 23728431347335, 20688443967788245, 23730591032609929084, 34687456062438088435890, 62994291032837018079196115, 139227352512368728514134480110, 368132634640135991872548754745138, 1147827724811251389730308940150980661

Offset: 2

Author

Michel Lagneau, May 14 2010

Keywords

• nonn
• easy

Comments

This is the case p=3 of a(n,p) = binomial(n^p,n) / ( PHI(n,p) * n^(p-1)) where PHI(n,p) = 1 + n + n^2 + ... + n^(p-1) = (n^p - 1) /(n - 1).

These a(n,p) are integer if n, p > = 2. [Proof :

a(n,p) = binomial(n^p,n)* 1 / (n^(p-1)*PHI(n,p))

= n^p *(n^p - 1)*(n^p - 2)...(n^p - n + 1)/((n-2)!*(n-1)*n * n^(p-1)* PHI(n,p)).

Insert PHI(n,p)=(n^p - 1) /(n - 1) and cancel n^p, n-1 and n^p - 1 where n > = 2:

a(n,p) = (n^p - 2)*(n^p - 3)...(n^p - n + 1)/(n - 2)! = binomial (n^p - 2, n - 2). QED]

Examples

			a(3) = binomial(3^3-2, 3-2) = binomial(25, 1) = 25.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 2..195

Crossrefs

Cf. A177234, A177784 (case p = 2).

Programs

Magma

[Binomial(n^3-2, n-2): n in [2..30]]; // G. C. Greubel, Jul 18 2024

Maple

with(numtheory): n0:=30: T:=array(1..n0): T:=array(1..n0-1):
for n from 2 to n0 do: p:=3: T[n-1]:= (n-1)*(binomial(n^p,n))/((n^(p-1))*(n^p-1)): od: print(T):

Mathematica

Table[Binomial[n^3-2,n-2],{n,2,20}] (* Harvey P. Dale, Aug 08 2013 *)

SageMath

[binomial(n^3-2,n-2) for n in range(2,31)] # G. C. Greubel, Jul 18 2024

Formula

a(n) = binomial(n^3, n) / (n^2 * (n^2 + n + 1) ).

Extensions

Swapped general and specific definitions - R. J. Mathar, Oct 12 2010