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A066796
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Sum(i=1,n,binomial(2*i,i)).
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24
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2, 8, 28, 98, 350, 1274, 4706, 17576, 66196, 250952, 956384, 3660540, 14061140, 54177740, 209295260, 810375650, 3143981870, 12219117170, 47564380970, 185410909790, 723668784230, 2827767747950, 11061198475550, 43308802158650
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Comments from Alexander Adamchuk (alex(AT)kolmogorov.com), 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 6n-1. A007528.
a(p-1) 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((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.
Every a(n) from a((p^2-1)/2) to a(p^2-1) is divisible by prime p>3.
a(p^2-1), a(p^2-2) and a(p^2-3) are divisible by p^2 for prime p>3.
a(p^2-4) is divisible by p^2 for prime p>5.
a(p^2-5) is divisible by p^2 for prime p>7.
a(p^2-6) is divisible by p^2 for prime p>7.
a(p^2-7) is divisible by p^2 for prime p>11.
a(p^2-8) 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 6n-1. A007528.
a(p^3-1) is divisible by p^2 for prime p=7,13.. Primes of form 6n+1. A002476.
a(p^4-1) 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(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 (alex(AT)kolmogorov.com), Jan 04 2007
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LINKS
| Harry J. Smith, Table of n, a(n) for n=1,...,200
Eric Weisstein's World of Mathematics, Central Binomial Coefficient.
Eric Weisstein's World of Mathematics, Binomial Sums.
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FORMULA
| a(n)=A006134(n)-1; generating function: (sqrt(1-4*x)-1)/(sqrt(1-4*x)*x*(x-1)) - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 11 2003
a(n) = Sum[ (2k)!/(k!)^2, {k,1,n} ]. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 02 2006
a(n) = Sum[ Binomial[2k,k], {k,1,n} ]. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jan 04 2007
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MATHEMATICA
| Table[Sum[(2k)!/(k!)^2, {k, 1, n}], {n, 1, 50}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 02 2006
Table[Sum[Binomial[2k, k], {k, 1, n}], {n, 1, 30}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Jan 04 2007
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PROG
| (PARI) { a=0; for (n=1, 200, write("b066796.txt", n, " ", a+=binomial(2*n, n)) ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Mar 27 2010]
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CROSSREFS
| Essentially the same as A079309 and A054114.
Cf. A006134, A079309, A083096, A007528, A002476.
Sequence in context: A090426 A060995 A106731 * A104934 A056711 A114590
Adjacent sequences: A066793 A066794 A066795 * A066797 A066798 A066799
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KEYWORD
| nonn
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AUTHOR
| Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 18 2002
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