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A014062 a(n) = binomial(n^2, n). 39
1, 1, 6, 84, 1820, 53130, 1947792, 85900584, 4426165368, 260887834350, 17310309456440, 1276749965026536, 103619293824707388, 9176358300744339432, 880530516383349192480, 91005567811177478095440 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

H. 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.

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

H. Alzer, J. Sandor, On a binomial coefficient and a product of prime numbers, Appl. An. Disc. Math. 5 (2011) 87-92.

H. J. Brothers, Pascal's Prism: Supplementary Material.

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

MATHEMATICA

Table[Binomial[n^2, n], {n, 0, 22}] (* Vladimir Joseph Stephan Orlovsky, Mar 03 2011 *)

PROG

(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

CROSSREFS

Cf. A295773.

Sequence in context: A277304 A128575 A322518 * A147626 A123312 A010794

Adjacent sequences: A014059 A014060 A014061 * A014063 A014064 A014065

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 7 13:02 EST 2022. Contains 358656 sequences. (Running on oeis4.)