A177454 ( binomial(2*p,p) - 2)/p where p = prime(n).

2, 6, 50, 490, 64130, 800046, 137270954, 1860277042, 357975249026, 1036802293087622, 15013817846943906, 47192717955016924590, 10360599532897359064118, 154361699651715243559786

Offset: 1

Author

Michel Lagneau, May 09 2010

Keywords

• nonn

Comments

All entries are integer because binomial(2p, p) == 2 (mod p). [Proof: p!*binomial(2p, p) = 2p(2p - 1)(2p - 2) ... (p + 1) .

Therefore (p - 1)!*binomial(2p, p) = 2(2p - 1) ... (p + 1) == 2(p - 1)! (mod p).

Since p is prime: (p - 1)! <> 0 (mod p). Because Z/pZ is a finite field, we conclude that binomial(2p, p) == 2 (mod p).]

Examples

			a(1) = 2 because prime(1) = 2 and (binomial(4, 2) - 2)/2 = (6 - 2)/2 = 2.
a(4) = 490 because prime(4) = 7 and (binomial(14, 7) - 2)/7 = (3432 - 2)/7 = 490.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..263
• Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, J. Integer Sequ., Vol. 9 (2006), Article 06.2.4.
• Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, J. Integer Sequ., Vol. 8 (2005), Article 05.4.5.

Crossrefs

Cf. A000984, A060842, A060545, A024498.

Programs

Magma

[(Binomial(2*p,p)-2)/p where p is NthPrime(n):n in [1..14]]; // Marius A. Burtea, Aug 11 2019

Maple

with(numtheory): n0:=20: T:=array(1..n0): k:=1: for n from 1 to 72 do:if type(n,prime)=true then T[k]:= (binomial(2*n,n)-2)/n: k:=k+1: fi: od: print(T):

Mathematica

Table[(Binomial[2Prime[n], Prime[n]] - 2)/Prime[n], {n, 15}] (* Alonso del Arte, Feb 27 2013 *)

Formula

a(n) = (A000984(p) - 2) / p with p = A000040(n).