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A079309 a(n) = C(1,1) + C(3,2) + C(5,3) + ... + C(2*n-1,n). 16

%I

%S 1,4,14,49,175,637,2353,8788,33098,125476,478192,1830270,7030570,

%T 27088870,104647630,405187825,1571990935,6109558585,23782190485,

%U 92705454895,361834392115,1413883873975,5530599237775,21654401079325,84859704298201,332818970772253

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

%C 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

%C 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

%C The inverse Catalan transform yields A024495, shifted once left. - _R. J. Mathar_, Jul 07 2009

%C From _Paul Barry_, Mar 29 2010: (Start)

%C Hankel transform is A138341.

%C 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)

%C a(n) is the number of terms of A031443 not exceeding 4^n. - _Vladimir Shevelev_, Oct 01 2010

%H Vincenzo Librandi and Robert Israel, <a href="/A079309/b079309.txt">Table of n, a(n) for n = 1..1500</a> (terms 1..200 from Vincenzo Librandi).

%H A. Denise and R. Simion, <a href="http://dx.doi.org/10.1016/0012-365X(93)E0147-V">Two combinatorial statistics on Dyck paths</a>, Discrete Math., 137, 1995, 155-176.

%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]

%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.

%H R. Witula, <a href="https://doi.org/10.1515/dema-2013-0418">Ramanujan type trigonometric formulas</a>, Demonstratio Mathematica, Vol. XLV, No. 4 (2012), 789-796. - From _N. J. A. Sloane_, Jan 01 2013

%F 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

%F G.f.: (1/sqrt(1 - 4*x) - 1)/(1 - x)/2. - _Vladeta Jovovic_, Feb 12 2003

%F Given g.f. A(x), then x * A(x - x^2) is g.f. of A024495. - _Michael Somos_, Feb 14 2006

%F a(n) = Sum_{j=1..n} binomial(2*j, j)/2. - _Zerinvary Lajos_, Oct 25 2006

%F a(n) = Sum_{0 <= i <= j <= n} binomial(i+j, i). - _Benoit Cloitre_, Nov 25 2006

%F 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

%F a(n) ~ 2^(2*n+1) / (3*sqrt(Pi*n)). - _Vaclav Kotesovec_, Feb 13 2014

%F a(n) = Sum_{k=0..n-1} A001700(k). - _Doug Bell_, Jun 23 2015

%F 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

%e a(4) = C(1,1) + C(3,2) + C(5,3) + C(7,4) = 1 + 3 + 10 + 35 = 49.

%e G.f. = x + 4*x^2 + 14*x^3 + 49*x^4 + 175*x^5 + 637*x^6 + 2353*x^7 + ...

%p a := n -> add(binomial(2*j, j)/2, j=1..n): seq(a(n), n=1..24); # _Zerinvary Lajos_, Oct 25 2006

%p a := n -> add(abs(binomial(-j, -2*j)), j=1..n): seq(a(n), n=1..24); # _Zerinvary Lajos_, Oct 03 2007

%p 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):

%p map(f, [$1..100]); # _Robert Israel_, Jun 24 2015

%t Rest[CoefficientList[Series[(1/Sqrt[1-4*x]-1)/(1-x)/2, {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Feb 13 2014 *)

%t Accumulate[Table[Binomial[2n-1,n],{n,30}]] (* _Harvey P. Dale_, Jan 06 2021 *)

%o (PARI) {a(n) = sum(k=1, n, binomial(2*k - 1, k))}; /* _Michael Somos_, Feb 14 2006 */

%o (PARI) x='x+O('x^100); Vec((1/sqrt(1-4*x)-1)/(1-x)/2) \\ _Altug Alkan_, Dec 24 2015

%Y Cf. A001700, A024495, A066796, A138341.

%Y Equals A024718(n) - 1.

%K easy,nonn

%O 1,2

%A _Miklos Kristof_, Feb 10 2003

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

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Last modified April 23 05:11 EDT 2021. Contains 343199 sequences. (Running on oeis4.)