

A066796


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


26



2, 8, 28, 98, 350, 1274, 4706, 17576, 66196, 250952, 956384, 3660540, 14061140, 54177740, 209295260, 810375650, 3143981870, 12219117170, 47564380970, 185410909790, 723668784230, 2827767747950, 11061198475550, 43308802158650
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OFFSET

1,1


COMMENTS

Comments from Alexander Adamchuk, Jul 02 2006: (Start)
Every a(n) is divisible by prime 2, a(n)/2 = A079309(n).
a(n) is divisible by prime 3 only for n=12,30,36,84,90,108,120,... A083096.
a(p) is divisible by p^2 for primes p=5,11,17,23,29,41,47,... Primes of form 6n1. A007528.
a(p1) is divisible by p^2 for primes p=7,13,19,31,37,43,... Primes of form 6n+1. A002476.
Every a(n) from a((p1)/2) to a(p1) is divisible by prime p for p=7,13,19,31,37,43,... Primes of form 6n+1. A002476.
Every a(n) from a((p^21)/2) to a(p^21) is divisible by prime p>3.
a(p^21), a(p^22) and a(p^23) are divisible by p^2 for prime p>3.
a(p^24) is divisible by p^2 for prime p>5.
a(p^25) is divisible by p^2 for prime p>7.
a(p^26) is divisible by p^2 for prime p>7.
a(p^27) is divisible by p^2 for prime p>11.
a(p^28) is divisible by p^2 for prime p>13.
a(p^3) is divisible by p^2 for prime 2 and prime p=5,11,... Primes of form 6n1. A007528.
a(p^31) is divisible by p^2 for prime p=7,13,... Primes of form 6n+1. A002476.
a(p^41) is divisible by p^2 for prime p>3. (End)
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(p1)/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^21) is divisible by p^2 for prime p>3. Every a(n) from a((4p+3)(p1)/6) to a((2p+3)(p1)/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


LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000 (Terms 1 to 200 computed by Harry J. Smith; terms 201 to 1000 computed by G. C. Greubel, Jan 15 2017)
GuoShuai Mao, Proof of a conjecture of Adamchuk, arXiv:2003.09810 [math.NT], 2020.
GuoShuai Mao, On a supercongruence conjecture of Z.W. Sun, arXiv:2003.14221 [math.NT], 2020.
GuoShuai Mao, Roberto Tauraso, Three pairs of congruences concerning sums of central binomial coefficients, arXiv:2004.09155 [math.NT], 2020.
Z.W. Sun, Fibonacci numbers modulo cubes of primes, arXiv:0911.3060 [math.NT], 20092013; Taiwanese J. Math., to appear 2013.  From N. J. A. Sloane, Mar 01 2013
Eric Weisstein's World of Mathematics, Central Binomial Coefficient.
Eric Weisstein's World of Mathematics, Binomial Sums.


FORMULA

a(n) = A006134(n)  1; generating function: (sqrt(14*x)1)/(sqrt(14*x)*(x1))  Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 11 2003, corrected by Vaclav Kotesovec, Nov 06 2012
a(n) = Sum_{k=1..n}(2k)!/(k!)^2.  Alexander Adamchuk, Jul 02 2006
a(n) = Sum_{k=1..n}binomial(2k,k).  Alexander Adamchuk, Jan 04 2007
a(n) ~ 2^(2*n+2)/(3*sqrt(Pi*n)).  Vaclav Kotesovec, Nov 06 2012


MATHEMATICA

Table[Sum[(2k)!/(k!)^2, {k, 1, n}], {n, 1, 50}] (* Alexander Adamchuk, Jul 02 2006 *)
Table[Sum[Binomial[2k, k], {k, 1, n}], {n, 1, 30}] (* Alexander Adamchuk, Jan 04 2007 *)


PROG

(PARI) { a=0; for (n=1, 200, write("b066796.txt", n, " ", a+=binomial(2*n, n)) ) } \\ Harry J. Smith, Mar 27 2010
(PARI) a(n) = sum(i=1, n, binomial(2*i, i)); \\ Michel Marcus, Jan 04 2016


CROSSREFS

Essentially the same as A079309 and A054114.
Equals A006134  1.
Cf. A002476, A006134, A007528, A079309, A083096.
Sequence in context: A318010 A291383 A277653 * A104934 A056711 A114590
Adjacent sequences: A066793 A066794 A066795 * A066797 A066798 A066799


KEYWORD

nonn


AUTHOR

Benoit Cloitre, Jan 18 2002


STATUS

approved



