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A079309
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a(n) = C(1,1) + C(3,2) + C(5,3) + ... + C(2*n-1,n).
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41
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1, 4, 14, 49, 175, 637, 2353, 8788, 33098, 125476, 478192, 1830270, 7030570, 27088870, 104647630, 405187825, 1571990935, 6109558585, 23782190485, 92705454895, 361834392115, 1413883873975, 5530599237775, 21654401079325, 84859704298201, 332818970772253
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OFFSET
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1,2
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COMMENTS
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a(n) is the sum of pyramid weights of all Dyck paths of length 2n (for pyramid weight see Denise and Simion). Equivalently, a(n) is the sum of the total lengths of end branches of an ordered tree, summation being over all ordered trees with n edges. For example, the five ordered trees with 3 edges have total lengths of endbranches 3,2,3,3 and 3. - Emeric Deutsch, May 30 2003
a(n) is the number of Motzkin paths of length 2n with exactly one level segment. (A level segment is a maximal sequence of contiguous flatsteps.) Example: for n=2, the paths counted are FFFF, FFUD, UDFF, UFFD. The formula for a(n) below counts these paths by length of the level segment. - David Callan, Jul 15 2004
The aerated sequence 0, 0, 1, 0, 4, 0, 14, 0, 49, ... has e.g.f. int(cosh(x-t)*Bessel_I(1,2t), t = 0..x). (End)
Also the number of nonempty subsets of {1..2n} with median n, bisection of A361801. The version containing n is A001700 (bisected). Replacing 2n with 2n+1 and n with n+1 gives A006134. For mean instead of median we have A212352. - Gus Wiseman, Apr 16 2023
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LINKS
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FORMULA
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a(n) = Sum_{0 <= i <= j <= n} binomial(i+j, i). - Benoit Cloitre, Nov 25 2006
D-finite with recurrence n*a(n) + (-5*n+2)*a(n-1) + 2*(2*n-1)*a(n-2) = 0. - R. J. Mathar, Nov 30 2012
a(n) = -binomial(2*n+1, n)*hypergeom([1, n+3/2], [n+2], 4) - (i/sqrt(3) + 1)/2. - Peter Luschny, May 18 2018
(End)
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EXAMPLE
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a(4) = C(1,1) + C(3,2) + C(5,3) + C(7,4) = 1 + 3 + 10 + 35 = 49.
G.f. = x + 4*x^2 + 14*x^3 + 49*x^4 + 175*x^5 + 637*x^6 + 2353*x^7 + ...
The a(1) = 1 through a(3) = 14 subsets of {1..2n} with median n:
{1} {2} {3}
{1,3} {1,5}
{1,2,3} {2,4}
{1,2,4} {1,3,4}
{1,3,5}
{1,3,6}
{2,3,4}
{2,3,5}
{2,3,6}
{1,2,4,5}
{1,2,4,6}
{1,2,3,4,5}
{1,2,3,4,6}
{1,2,3,5,6}
(End)
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MAPLE
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a := n -> add(binomial(2*j, j)/2, j=1..n): seq(a(n), n=1..24); # Zerinvary Lajos, Oct 25 2006
a := n -> add(abs(binomial(-j, -2*j)), j=1..n): seq(a(n), n=1..24); # Zerinvary Lajos, Oct 03 2007
f:= gfun:-rectoproc({n*a(n) +(-5*n+2)*a(n-1) +2*(2*n-1)*a(n-2)=0, a(1)=1, a(2)=4}, a(n), remember):
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MATHEMATICA
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Rest[CoefficientList[Series[(1/Sqrt[1-4*x]-1)/(1-x)/2, {x, 0, 20}], x]] (* Vaclav Kotesovec, Feb 13 2014 *)
Accumulate[Table[Binomial[2n-1, n], {n, 30}]] (* Harvey P. Dale, Jan 06 2021 *)
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PROG
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(PARI) {a(n) = sum(k=1, n, binomial(2*k - 1, k))}; /* Michael Somos, Feb 14 2006 */
(PARI) x='x+O('x^100); Vec((1/sqrt(1-4*x)-1)/(1-x)/2) \\ Altug Alkan, Dec 24 2015
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CROSSREFS
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This is the even (or odd) bisection of A361801.
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 11 2003
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STATUS
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approved
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