A177837 Binomial(n^3,n) / (n^2 * (n^2+n+1) ), or binomial(n^3-2,n-2).

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

Offset

2,2

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]

Links

Table of n, a(n) for n=2..13.

Example

a(n= 3) = binomial(27,3)/(9 *(9 + 3 + 1)= 2925 /117 = 25 = binomial(3^3 - 2, 3 - 2) = binomial (25, 1).

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 *)

Crossrefs

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

Sequence in context: A172261 A023113 A322247 * A056047 A281436 A197671

Adjacent sequences: A177834 A177835 A177836 * A177838 A177839 A177840

Keyword

nonn

Author

Michel Lagneau, May 14 2010

Extensions

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

Status

approved