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A177837 Binomial(n^3,n) / (n^2 * (n^2+n+1) ), or binomial(n^3-2,n-2). 0

%I

%S 1,25,1891,302621,84957251,37307689133,23728431347335,

%T 20688443967788245,23730591032609929084,34687456062438088435890,

%U 62994291032837018079196115,139227352512368728514134480110

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

%C 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).

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

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

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

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

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

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

%p with(numtheory): n0:=30: T:=array(1..n0): T:=array(1..n0-1):

%p 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):

%t Table[Binomial[n^3-2,n-2],{n,2,20}] (* _Harvey P. Dale_, Aug 08 2013 *)

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

%K nonn

%O 2,2

%A _Michel Lagneau_, May 14 2010

%E Swapped general and specific definitions - _R. J. Mathar_, Oct 12 2010

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Last modified August 17 15:35 EDT 2022. Contains 356189 sequences. (Running on oeis4.)