A112861 -id:A112861 - cp's OEIS frontend

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

N. J. A. Sloane

(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

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.

Cf. A001700, A001701.

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

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

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

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 *)

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

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)

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)

A066726 Numbers n such that binomial(2n, n) - 1 is prime.

Original entry on oeis.org

2, 3, 5, 9, 15, 29, 43, 51, 113, 184, 213, 222, 267, 279, 369, 402, 441, 603, 812, 839, 902, 1422, 1542, 1824, 2983, 3065, 3911, 3958, 4192, 4587, 4865, 5543, 5837, 7902, 9299, 9722, 10412, 10648, 11498, 12803, 14428, 15876, 20173, 26311, 38927, 52210, 54189, 59757, 60454, 72094, 76899, 85033, 91059, 91059

Offset: 1

Robert G. Wilson v, Jan 15 2002

nonn

I.e., numbers n such that (2*n)!/(n!)^2-1 is prime. - Hugo Pfoertner, Sep 25 2005
The next term is > 30000. - Vaclav Kotesovec, May 03 2021
a(55) > 100000. - Robert Price, Jul 02 2024

Cf. A066699, A085793.

Cf. A092751 = primes of the form (2*n)!/(n!)^2-1, A112853 = (2*n)!/n!-1 is prime, A112855 = (2*n)!/n!+1 is prime, A066699 = (2*n)!/(n!)^2+1 is prime, A112861 = (2*n)!/(2*(n!)^2)-1 is prime, A112863 = (2*n)!/(2*(n!)^2)+1 is prime. - Hugo Pfoertner, Sep 25 2005

Do[ If[ PrimeQ[ Binomial[2n, n] - 1], Print[n]], {n, 1, 2000} ]

is(n)=isprime(binomial(2*n,n)-1) \\ Charles R Greathouse IV, Feb 17 2017

More terms from Ed Pegg Jr, Sep 10 2003
Edited by N. J. A. Sloane, Aug 23 2008 at the suggestion of R. J. Mathar
a(43)-a(44) from Vaclav Kotesovec, May 03 2021
a(45)-a(54) from Robert Price, Jul 02 2024

A112853 Numbers k such that (2*k)!/k!-1 is prime.

Original entry on oeis.org

2, 6, 14, 25, 114, 188, 193, 361, 712, 1767, 3195, 4995, 5682, 6485, 9937

Offset: 1

Hugo Pfoertner, Sep 25 2005

Next term is > 9680.

Cf. A112854 primes of the form (2*n)!/n!-1, A112855 (2*n)!/n!+1 is prime, A066726 (2*n)!/(n!)^2-1 is prime, A066699 (2*n)!/(n!)^2+1 is prime, A112861 (2*n)!/(2*(n!)^2)-1 is prime, A112863 (2*n)!/(2*(n!)^2)+1 is prime.

Select[Range[7000], PrimeQ[(2#)!/#!-1]&] (* James C. McMahon, Jun 12 2024 *)

a(15) from Michael S. Branicky, Jun 12 2024

A112855 Numbers n such that (2*n)!/n!+1 is prime.

Original entry on oeis.org

1, 2, 5, 10, 24, 30, 72, 340, 379, 712, 1020, 1647, 3654, 7923

Offset: 1

Hugo Pfoertner, Sep 25 2005

Next term is > 10000.

Cf. A112856 primes of the form (2*n)!/n!+1, A112853 (2*n)!/n!-1 is prime, A066726 (2*n)!/(n!)^2-1 is prime, A112859 (2*n)!/(n!)^2+1 is prime, A112861 (2*n)!/(2*(n!)^2)-1 is prime, A112863 (2*n)!/(2*(n!)^2)+1 is prime.

Select[Range[8000],PrimeQ[(2#)!/#!+1]&] (* Harvey P. Dale, Mar 23 2012 *)

A112863 Numbers k such that (2k)!/(2(k!)^2)+1 is prime.

Original entry on oeis.org

1, 3, 5, 6, 14, 19, 24, 45, 49, 62, 72, 122, 149, 197, 209, 222, 349, 409, 491, 527, 601, 675, 728, 769, 795, 853, 1039, 1318, 1599, 2069, 2279, 2567, 2634, 3286, 3482, 6880, 6919, 9117, 9276, 9564, 10898, 13061, 14570, 16493, 17366, 18773, 19901, 25569, 26611, 28322

Offset: 1

Hugo Pfoertner, Sep 30 2005

Cf. A001700, A112864, A112853, A112855, A066726, A112861.

Select[Range[3000],PrimeQ[(2#)!/(2(#!)^2)+1]&] (* James C. McMahon, Jun 13 2024 *)

isok(k) = ispseudoprime(1+binomial(2*k-1, k-1)); \\ Michel Marcus, Jun 14 2024

a(42)-a(43) from Charles R Greathouse IV, Sep 26 2006

a(44)-a(50) from Michael S. Branicky, Jul 24 2024

A112862 Primes of the form (2k)!/(2(k!)^2)-1.

Original entry on oeis.org

2, 461, 92377, 269128937219, 51913710643776705684835559, 3017467217880703353213932318284163999

Offset: 1

Hugo Pfoertner, Sep 30 2005

nonn

The next term (2*306)!/(2*(306!)^2)-1 has 183 decimal digits.

Cf. A001700, A112861, A112864.

Select[Table[(2n)!/(2(n!)^2)-1, {n, 70}],PrimeQ] (* James C. McMahon, Jun 13 2024 *)

cp's OEIS Frontend

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

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A066726 Numbers n such that binomial(2n, n) - 1 is prime.

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A112853 Numbers k such that (2*k)!/k!-1 is prime.

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A112855 Numbers n such that (2*n)!/n!+1 is prime.

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A112863 Numbers k such that (2*k)!/(2*(k!)^2)+1 is prime.

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A112862 Primes of the form (2*k)!/(2*(k!)^2)-1.

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A112863 Numbers k such that (2k)!/(2(k!)^2)+1 is prime.

A112862 Primes of the form (2k)!/(2(k!)^2)-1.