A010763 a(n) = binomial(2n+1, n+1) - 1.

0, 2, 9, 34, 125, 461, 1715, 6434, 24309, 92377, 352715, 1352077, 5200299, 20058299, 77558759, 300540194, 1166803109, 4537567649, 17672631899, 68923264409, 269128937219, 1052049481859, 4116715363799, 16123801841549, 63205303218875, 247959266474051

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

(With a different offset:) p divides a(p) for prime p. p^2 divides a(p) for prime p > 2. p^3 divides a(p) for prime p > 3 (implied by Wolstenholme's theorem). Wolstenholme's quotients are listed in A034602(n) = a(prime(n))/prime(n)^3 = {1, 5, 265, 2367, 237493, 2576561, 338350897, ...} = a(p)/p^3 for prime p > 3. p^3 divides a(p^k) for prime p > 3 and integer k > 0. Primes in a(n) are listed in A112862(n) = {2, 461, 92377, 269128937219, ...} Primes of the form (2*n)!/(2*(n!)^2) - 1. Numbers n such that a(n) is prime are listed in A112861(n) = {2, 6, 10, 21, 45, 63, 306, 404, 437, 471, 646, ...}. - Alexander Adamchuk, Jan 05 2007

a(n-1) is the number of weak compositions of n into n parts in which at least one part is zero. a(3)=34 since 4 can be written as 4+0+0+0 (4 such compositions); 3+1+0+0 (12 such compositions); 2+2+0+0 (6 such compositions); 2+1+1+0 (12 such compositions). All these weak compositions contain at least one zero. - Enrique Navarrete, Jan 09 2022

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Wolstenholme's Theorem
• Jianqiang Zhao, Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota-Baxter Algebras, arXiv preprint arXiv:1412.8044 [math.NT], 2014.

Crossrefs

Cf. A001700, A001701.

Cf. A001008, A007406, A112861, A112862, A034602.

Programs

Magma

[Binomial(2*n-1,n-1)-1: n in [1..30]]; // Vincenzo Librandi, Mar 21 2013

Maple

A010763:=n->binomial(2*n+1, n+1) - 1: seq(A010763(n), n=0..30); # Wesley Ivan Hurt, Sep 05 2015

Mathematica

Table[Binomial[2n - 1, n - 1] - 1, {n, 20}] (* Alonso del Arte, Dec 13 2012 *)
CoefficientList[Series[Exp[2*x]*(BesselI[0,2*x] + BesselI[1,2*x]) - Exp[x], {x, 0, 20}], x]*Table[n!, {n, 0, 20}] (* Stefano Spezia, Dec 02 2018 *)

PARI

a(n) = binomial(2*n+1, n+1) - 1;
vector(30, n, a(n-1)) \\ Michel Marcus, Sep 05 2015

PARI

first(n) = x='x+O('x^n); Vec((1 - sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)) - 1/(1 - x), -n) \\ Iain Fox, Dec 19 2017 (corrected by Iain Fox, Oct 24 2018)

Formula

a(n) = (n/(2n+2))*Sum_{k = 1..n+1} C(2n+2, k)/C(n+1, k). - Benoit Cloitre, Aug 20 2002

a(n) = Sum_{i = 1..n} C(n + i, n). - Benoit Cloitre, Oct 15 2002

a(n + 1) = C(2n - 1, n - 1) - 1. - Alonso del Arte, Dec 15 2012

From Ilya Gutkovskiy, Feb 07 2017: (Start)

O.g.f.: (1 - sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)) - 1/(1 - x).

E.g.f.: exp(2*x)*(BesselI(0,2*x) + BesselI(1,2*x)) - exp(x). (End)