This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A177837 #21 Oct 14 2024 00:11:01
%S A177837 1,25,1891,302621,84957251,37307689133,23728431347335,
%T A177837 20688443967788245,23730591032609929084,34687456062438088435890,
%U A177837 62994291032837018079196115,139227352512368728514134480110,368132634640135991872548754745138,1147827724811251389730308940150980661
%N A177837 a(n) = binomial(n^3-2, n-2).
%C A177837 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 A177837 These a(n,p) are integer if n, p > = 2. [Proof :
%C A177837 a(n,p) = binomial(n^p,n)* 1 / (n^(p-1)*PHI(n,p))
%C A177837 = n^p *(n^p - 1)*(n^p - 2)...(n^p - n + 1)/((n-2)!*(n-1)*n * n^(p-1)* PHI(n,p)).
%C A177837 Insert PHI(n,p)=(n^p - 1) /(n - 1) and cancel n^p, n-1 and n^p - 1 where n > = 2:
%C A177837 a(n,p) = (n^p - 2)*(n^p - 3)...(n^p - n + 1)/(n - 2)! = binomial (n^p - 2, n - 2). QED]
%H A177837 G. C. Greubel, <a href="/A177837/b177837.txt">Table of n, a(n) for n = 2..195</a>
%F A177837 a(n) = binomial(n^3, n) / (n^2 * (n^2 + n + 1) ).
%e A177837 a(3) = binomial(3^3-2, 3-2) = binomial(25, 1) = 25.
%p A177837 with(numtheory): n0:=30: T:=array(1..n0): T:=array(1..n0-1):
%p A177837 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 A177837 Table[Binomial[n^3-2,n-2],{n,2,20}] (* _Harvey P. Dale_, Aug 08 2013 *)
%o A177837 (Magma) [Binomial(n^3-2, n-2): n in [2..30]]; // _G. C. Greubel_, Jul 18 2024
%o A177837 (SageMath) [binomial(n^3-2,n-2) for n in range(2,31)] # _G. C. Greubel_, Jul 18 2024
%Y A177837 Cf. A177234, A177784 (case p = 2).
%K A177837 nonn,easy
%O A177837 2,2
%A A177837 _Michel Lagneau_, May 14 2010
%E A177837 Swapped general and specific definitions - _R. J. Mathar_, Oct 12 2010