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A177837 Binomial(n^3,n) / (n^2 * (n^2+n+1) ), or binomial(n^3-2,n-2). 0
1, 25, 1891, 302621, 84957251, 37307689133, 23728431347335, 20688443967788245, 23730591032609929084, 34687456062438088435890, 62994291032837018079196115, 139227352512368728514134480110 (list; graph; refs; listen; history; text; internal format)
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

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Last modified February 21 15:31 EST 2019. Contains 320374 sequences. (Running on oeis4.)