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A079309 a(n) = C(1,1) + C(3,2) + C(5,3) + ... + C(2n-1,n). 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

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

Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]

R. Witula, Ramanujan type trigonometric formulas, Demonstratio Mathematica, Vol. XLV, No. 4, 2012, pp. 789-796. - From N. J. A. Sloane, Jan 01 2013

FORMULA

a(n) = (1/2)*(C(2, 1) + C(4, 2) + C(6, 3) + ... + C(2n, 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) = Sum_{j=1..n} binomial(2*j,j)/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

EXAMPLE

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 + ...

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 *)

PROG

(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

CROSSREFS

Cf. A001700, A024495, A066796, A138341.

Equals A024718(n) - 1.

Sequence in context: A278026 A001894 A215493 * A026630 A034459 A120747

Adjacent sequences:  A079306 A079307 A079308 * A079310 A079311 A079312

KEYWORD

easy,nonn

AUTHOR

Miklos Kristof, Feb 10 2003

EXTENSIONS

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

STATUS

approved

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Last modified April 5 20:19 EDT 2020. Contains 333260 sequences. (Running on oeis4.)