A079309 a(n) = C(1,1) + C(3,2) + C(5,3) + ... + C(2*n-1,n).

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

Offset: 1

Author

Miklos Kristof, Feb 10 2003

Keywords

• easy
• nonn

Comments

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 inverse Catalan transform yields A024495, shifted once left. - R. J. Mathar, Jul 07 2009

From Paul Barry, Mar 29 2010: (Start)

Hankel transform is A138341.

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)

a(n) is the number of terms of A031443 not exceeding 4^n. - Vladimir Shevelev, Oct 01 2010

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

Examples

			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 + ...
From _Gus Wiseman_, Apr 16 2023: (Start)
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)
		

Links

• Vincenzo Librandi and Robert Israel, Table of n, a(n) for n = 1..1500 (terms 1..200 from Vincenzo Librandi).
• A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176.
• Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
• Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
• R. Witula, Ramanujan type trigonometric formulas, Demonstratio Mathematica, Vol. XLV, No. 4 (2012), 789-796. - From _N. J. A. Sloane_, Jan 01 2013

Crossrefs

Cf. A001700, A024495, A066796, A138341.

Equals A024718(n) - 1.

This is the even (or odd) bisection of A361801.

A007318 counts subsets by length, A327481 by mean, A013580 by median.

A359893 and A359901 count partitions by median.

Cf. A000975, A057552, A231147, A361654, A361864, A362046.

Programs

Maple

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):
map(f, [$1..100]); # Robert Israel, Jun 24 2015

Mathematica

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 *)

PARI

{a(n) = sum(k=1, n, binomial(2*k - 1, k))}; /* Michael Somos, Feb 14 2006 */

PARI

my(x='x+O('x^40)); Vec((1/sqrt(1-4*x)-1)/(1-x)/2) \\ Altug Alkan, Dec 24 2015

Formula

a(n) = (1/2)*(C(2, 1) + C(4, 2) + C(6, 3) + ... + C(2*n, n)) = A066796(n)/2. - Vladeta Jovovic, Feb 12 2003

G.f.: (1/sqrt(1 - 4*x) - 1)/(1 - x)/2. - Vladeta Jovovic, Feb 12 2003

Given g.f. A(x), then x * A(x - x^2) is g.f. of A024495. - Michael Somos, Feb 14 2006

a(n) = A066796(n)/2. - Zerinvary Lajos, Oct 25 2006

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) ~ 2^(2*n+1) / (3*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 13 2014

a(n) = Sum_{k=0..n-1} A001700(k). - Doug Bell, Jun 23 2015

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

From Gus Wiseman, Apr 18 2023: (Start)

a(n) = A024718(n) - 1.

a(n) = A231147(2n+1,n).

a(n) = A361801(2n) = A361801(2n+1). (End)

a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(2*n+2-k, n-2*k). - Michael Weselcouch, Jun 17 2025

a(n) = binomial(2*(1+n), n)*hypergeom([1, (1-n)/2, -n/2], [-2*(1+n), 3+n], 4). - Stefano Spezia, Jun 18 2025

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 11 2003