A066796 - OEIS

%I #67 Feb 16 2025 08:32:45

%S 2,8,28,98,350,1274,4706,17576,66196,250952,956384,3660540,14061140,

%T 54177740,209295260,810375650,3143981870,12219117170,47564380970,

%U 185410909790,723668784230,2827767747950,11061198475550,43308802158650

%N a(n) = Sum_{i=1..n} binomial(2*i,i).

%C Comments from _Alexander Adamchuk_, Jul 02 2006: (Start)

%C Every a(n) is divisible by prime 2, a(n)/2 = A079309(n).

%C a(n) is divisible by prime 3 only for n=12,30,36,84,90,108,120,... A083096.

%C a(p) is divisible by p^2 for primes p=5,11,17,23,29,41,47,... Primes of form 6n-1. A007528.

%C a(p-1) is divisible by p^2 for primes p=7,13,19,31,37,43,... Primes of form 6n+1. A002476.

%C Every a(n) from a((p-1)/2) to a(p-1) is divisible by prime p for p=7,13,19,31,37,43,... Primes of form 6n+1. A002476.

%C Every a(n) from a((p^2-1)/2) to a(p^2-1) is divisible by prime p>3.

%C a(p^2-1), a(p^2-2) and a(p^2-3) are divisible by p^2 for prime p>3.

%C a(p^2-4) is divisible by p^2 for prime p>5.

%C a(p^2-5) is divisible by p^2 for prime p>7.

%C a(p^2-6) is divisible by p^2 for prime p>7.

%C a(p^2-7) is divisible by p^2 for prime p>11.

%C a(p^2-8) is divisible by p^2 for prime p>13.

%C a(p^3) is divisible by p^2 for prime 2 and prime p=5,11,... Primes of form 6n-1. A007528.

%C a(p^3-1) is divisible by p^2 for prime p=7,13,... Primes of form 6n+1. A002476.

%C a(p^4-1) is divisible by p^2 for prime p>3. (End)

%C Mod[ a(3^k), 9 ] = 1 for integer k>0. Smallest number k such that 2^n divides a(k) is k(n) = {1,2,2,11,11,46,46,707,707,707,...}. Smallest number k such that 3^n divides a(k) is k(n) = {12,822,2466,...}. a(2(p-1)/3) is divisible by p^2 for prime p = {7,13,19,31,37,43,61,...} = A002476 Primes of form 6n+1. Every a(n) from a(p^2-(p+1)/2) to a(p^2-1) is divisible by p^2 for prime p>3. Every a(n) from a((4p+3)(p-1)/6) to a((2p+3)(p-1)/3) is divisible by p^2 for prime p = {7,13,19,31,37,43,61,...} = A002476 Primes of form 6n+1. - _Alexander Adamchuk_, Jan 04 2007

%H G. C. Greubel, <a href="/A066796/b066796.txt">Table of n, a(n) for n = 1..1000</a> (Terms 1 to 200 computed by Harry J. Smith; terms 201 to 1000 computed by G. C. Greubel, Jan 15 2017)

%H Guo-Shuai Mao, <a href="https://arxiv.org/abs/2003.09810">Proof of a conjecture of Adamchuk</a>, arXiv:2003.09810 [math.NT], 2020.

%H Guo-Shuai Mao, <a href="https://arxiv.org/abs/2003.14221">On a supercongruence conjecture of Z.-W. Sun</a>, arXiv:2003.14221 [math.NT], 2020.

%H Guo-Shuai Mao, <a href="https://doi.org/10.13140/RG.2.2.34392.88324">On some supercongruence conjectures of Z.-W. Sun</a>, Nanjing Univ. Info. Sci. Tech. (China, 2023).

%H Guo-Shuai Mao, <a href="https://www.researchgate.net/profile/Guo-Shuai-Mao-2/publication/375525646_Proof_of_some_congruences_via_the_hypergeometric_identities/">Proof of some congruences via the hypergeometric identities</a>, Nanjing Univ. Info. Sci. Tech. (China, 2023).

%H Guo-Shuai Mao, <a href="https://doi.org/10.13140/RG.2.2.36651.96802">On two pairs of congruence conjectures of Z.-W. Sun</a>, Nanjing Univ. Sci. Tech. (China), ResearchGate, 2024.

%H Guo-Shuai Mao and Roberto Tauraso, <a href="https://arxiv.org/abs/2004.09155">Three pairs of congruences concerning sums of central binomial coefficients</a>, arXiv:2004.09155 [math.NT], 2020.

%H Guo-Shuai Mao and Dong-Hui Zhang, <a href="https://www.researchgate.net/profile/Guo-Shuai-Mao-2/publication/375381277_On_some_congruences_involving_harmonic_numbers_and_Bernoulli_polynomials/">On some congruences involving harmonic numbers and Bernoulli polynomials</a>, ResearchGate, 2023. See pp. 1-2, 12.

%H Z.-W. Sun, <a href="http://arxiv.org/abs/0911.3060">Fibonacci numbers modulo cubes of primes</a>, arXiv:0911.3060 [math.NT], 2009-2013; Taiwanese J. Math., to appear 2013. - From _N. J. A. Sloane_, Mar 01 2013

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CentralBinomialCoefficient.html">Central Binomial Coefficient</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BinomialSums.html">Binomial Sums</a>.

%F a(n) = A006134(n) - 1; generating function: (sqrt(1-4*x)-1)/(sqrt(1-4*x)*(x-1)) - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 11 2003, corrected by _Vaclav Kotesovec_, Nov 06 2012

%F a(n) = Sum_{k=1..n}(2k)!/(k!)^2. - _Alexander Adamchuk_, Jul 02 2006

%F a(n) = Sum_{k=1..n}binomial(2k,k). - _Alexander Adamchuk_, Jan 04 2007

%F a(n) ~ 2^(2*n+2)/(3*sqrt(Pi*n)). - _Vaclav Kotesovec_, Nov 06 2012

%t Table[Sum[(2k)!/(k!)^2,{k,1,n}],{n,1,50}] (* _Alexander Adamchuk_, Jul 02 2006 *)

%t Table[Sum[Binomial[2k,k],{k,1,n}],{n,1,30}] (* _Alexander Adamchuk_, Jan 04 2007 *)

%o (PARI) { a=0; for (n=1, 200, write("b066796.txt", n, " ", a+=binomial(2*n, n)) ) } \\ _Harry J. Smith_, Mar 27 2010

%o (PARI) a(n) = sum(i=1, n, binomial(2*i,i)); \\ _Michel Marcus_, Jan 04 2016

%Y Essentially the same as A079309 and A054114.

%Y Equals A006134 - 1.

%Y Cf. A002476, A006134, A007528, A079309, A083096.

%K nonn,changed

%O 1,1

%A _Benoit Cloitre_, Jan 18 2002