A177466 a(n) = binomial(n^3, n^2) / (n^2 + n + 1).

10, 360525, 23263187479980, 4195317468983232014706855, 3118254010126197540790713959812283024388, 13329519847131745416659896296893907619682838146506167497550

Offset: 2

Author

Michel Lagneau, May 09 2010

Keywords

• nonn

Comments

All entries are integers. [Proof: binomial(n^3, n^2) / (n^2 + n + 1) = n^3 (n^3 - 1) (n^3 - 2)*...*(n^3- n^2 +1) / ( (n^2)! *(n^2 + n + 1)). With n^3 - 1 = (n-1)*(n^2 + n + 1), we obtain a(n) = n* binomial(n^3-2, n^2-2) / (n+1). Finally: (n+1) * binomial(n^3, n^2) * 1/ (n^2 + n + 1) = n*binomial(n^3-2, n^2-2). QED]

The step after "finally" seems to demonstrate merely that (n+1)*a(n) is an integer, but not that a(n) is itself an integer. Is the proof incomplete? - R. J. Mathar, Dec 06 2010

So far all that has been shown is that (n+1)*a(n) is an integer. To complete the proof, note that a(n) = n^3*(n-1)*(n^3-2)*...*(n^3-n^2) / (n^2*(n^2-1)!*(n^3-n^2)) = binomial(n^3-2,n^2-1)/n. Hence n*a(n) is also an integer, and so (n+1)*a(n) - n*a(n) = a(n) is an integer. Q.E.D. - N. J. A. Sloane, Dec 09 2010

Examples

			For n = 2, a(2) = binomial(8,4)/7 = 70/7 = 10.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 2..23
• Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Crossrefs

Cf. A000108, A000984, A060545, A177234, A177456.

Programs

Magma

[Binomial(n^3,n^2)/(n^2+n+1): n in [2..12]]; // G. C. Greubel, Jul 18 2024

Maple

A177466 := proc(n) binomial(n^3,n^2)/(n^2+n+1); end proc:
seq(A177466(n),n=2..10) ; # R. J. Mathar, Dec 06 2010

Mathematica

Table[Binomial[n^3,n^2]/(n^2+n+1),{n,2,7}] (* Harvey P. Dale, Jan 24 2019 *)

SageMath

[binomial(n^3,n^2)/(n^2+n+1) for n in range(2,13)] # G. C. Greubel, Jul 18 2024