A177456 a(n) = binomial(n^2,n+1)/n.

2, 42, 1092, 35420, 1391280, 64425438, 3442573064, 208710267480, 14162980464360, 1063958304188780, 87677864005521636, 7865449972066576656, 763126447532235966816, 79629871834780293333510

Offset: 2

Author

Michel Lagneau, May 09 2010

Keywords

• nonn

Comments

n divides binomial(n^2,n+1).

Proof 1 :(n+1)*binomial(n^2,n+1) = n*(n-1)*binomial(n^2,n) => n divide binomial(n^2,n+1) because gcd(n,n+1) = 1.

Proof 2 : a(n) = binomial(n^2,n+1)/n = (n-1)*binomial(n^2-2,n-1)=> a(n) is an integer. - Michel Lagneau, May 13 2010

Examples

			For n=4, 1092 is in the sequence because binomial(16,5)/4 = 4368/4 = 1092.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 2..335
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
• Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Crossrefs

Cf. A000984, A014062, A060545, A177234, A177784, A177788.

Programs

Magma

[Binomial(n^2,n+1)/n: n in [2..30]]; // G. C. Greubel, Apr 29 2024

Maple

with(numtheory):n0:=30:T:=array(1..n0-1):for n from 2 to n0 do:T[n-1]:= (binomial(n*n,n+1))/n:od:print(T):

Mathematica

Table[Binomial[n^2,n+1]/n, {n,2,30}] (* G. C. Greubel, Apr 29 2024 *)

SageMath

[binomial(n^2,n+1)/n for n in range(2,31)] # G. C. Greubel, Apr 29 2024

Formula

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

From G. C. Greubel, Apr 29 2024: (Start)

a(n) = (n-1)*A177234(n).

a(n) = (n-1)*A177788(n)/n.

a(n) = n*(n-1)*A177784(n).

a(n) = A014062(n)/n. (End)