A014062 a(n) = binomial(n^2, n).

1, 1, 6, 84, 1820, 53130, 1947792, 85900584, 4426165368, 260887834350, 17310309456440, 1276749965026536, 103619293824707388, 9176358300744339432, 880530516383349192480, 91005567811177478095440, 10078751602022313874633200, 1190739044344491048895397910

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Roberts states that Gupta and Khare show that a(n) > A002110(n) for 2 < n < 1794 and that a(n) < A002110(n) for n >= 1794, where A002110(n) is the product of the first n primes. - T. D. Noe, Oct 03 2007

This sequence describes the number of ways to arrange n objects in an n X n array (for example, stars in a flag's field pattern). - Tom Young (mcgreg265(AT)msn.com), Jun 17 2010

It appears that a(n) == n (mod n^3) only if n is 1, an odd prime, the square of an odd prime, or the cube of an odd prime. - Gary Detlefs, Aug 06 2013; corrected by Michel Marcus, May 29 2015

References

• J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 265.

Links

• T. D. Noe, Table of n, a(n) for n=0..100
• Horst A. Alzer and Jozsef J. B. Šandor, On a binomial coefficient and a product of prime numbers, Appl. An. Disc. Math. 5 (2011) 87-92.
• Harlan J. Brothers, Pascal's Prism: Supplementary Material.
• Harlan J. Brothers, Pascal's triangle, Sidi polynomials, and powers of e, Missouri J. Math. Sci. (2025) Vol. 37, No. 1, 67-78.
• Hansraj Gupta and S. P. Khare, On C(k^2,k) and the product of the first k primes, Publ. Fac. Electrotechn. Belgrade, Ser. Math. Phys. 25-29 (1977) 577-598.

Crossrefs

Main diagonal of A060539.

Cf. A177234, A177456, A177784, A177788, A272094, A295773.

Programs

Magma

[Binomial(n^2,n): n in [0..30]]; // G. C. Greubel, Apr 29 2024

Mathematica

Table[Binomial[n^2,n],{n,0,22}] (* Vladimir Joseph Stephan Orlovsky, Mar 03 2011 *)
Table[SeriesCoefficient[(1+x)^(n^2), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 06 2025 *)

PARI

{a(n) = sum(k=0, n, binomial(n, k)*binomial(n^2-n, k))}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Nov 18 2015

SageMath

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

Formula

a(n) ~ 1/sqrt(2*Pi) * (e*n)^(n - 1/2). - Charles R Greathouse IV, Jul 07 2007

a(n) = Sum_{k=0..n} binomial(n, k) * binomial(n^2 - n, k). - Paul D. Hanna, Nov 18 2015

a(n) = (n+1)*A177234(n). - R. J. Mathar, Jan 25 2019

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

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

a(n) = (n+1)*A177456(n)/(n-1).

a(n) = (n+1)*A177788(n)/n. (End)

a(n) = [x^n] (1+x)^(n^2). - Vaclav Kotesovec, Aug 06 2025