A097613 a(n) = binomial(2n-3,n-1) + binomial(2n-2,n-2).

1, 2, 7, 25, 91, 336, 1254, 4719, 17875, 68068, 260338, 999362, 3848222, 14858000, 57500460, 222981435, 866262915, 3370764540, 13135064250, 51250632510, 200205672810, 782920544640, 3064665881940, 12007086477750, 47081501377326, 184753963255176, 725510446350004

Offset: 1

Author

David Callan, Sep 20 2004

Keywords

• nonn

Comments

a(n) is the number of Dyck (2n-1)-paths with maximum pyramid size = n. A pyramid in a Dyck path is a maximal subpath of the form k upsteps immediately followed by k downsteps and its size is k.

a(n) is the total number of runs of peaks in all Dyck (n+1)-paths. A run of peaks is a maximal subpath of the form (UD)^k with k>=1. For example, a(2)=7 because the 5 Dyck 3-paths contain a total of 7 runs of peaks (in uppercase type): uuUDdd, uUDUDd, uUDdUD, UDuUDd, UDUDUD. - David Callan, Jun 07 2006

Binomial transform of A113682. - Paul Barry, Aug 21 2007

If Y is a fixed 2-subset of a (2n+1)-set X then a(n+1) is the number of n-subsets of X intersecting Y. - Milan Janjic, Oct 21 2007

Equals the Catalan sequence, A000108, convolved with A051924 prefaced with a 1: (1, 1, 4, 14, 50, ...). - Gary W. Adamson, May 15 2009

Central terms of triangle A209561. - Reinhard Zumkeller, Dec 26 2012

Also the number of compositions of 2*(n-1) in which the odd parts appear as many times in odd as in even positions. - Alois P. Heinz, May 26 2018

Examples

			a(2) = 2 because UUDDUD and UDUUDD each have maximum pyramid size = 2.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..1664
• Paul Barry, Extensions of Riordan Arrays and Their Applications, Mathematics (2025) Vol. 13, No. 2, 242. See p. 16.
• Gi-Sang Cheon, Hana Kim, and Louis W. Shapiro, Mutation effects in ordered trees, arXiv preprint arXiv:1410.1249 [math.CO], 2014.
• Milan Janjic, Two Enumerative Functions
• Toufik Mansour and I. L. Ramirez, Enumerations of polyominoes determined by Fuss-Catalan words, Australas. J. Combin. 81 (3) (2021) 447-457, table 2.
• Lin Yang and Shengliang Yang, Protected Branches in Ordered Trees, J. Math. Study (2023) Vol. 56, No. 1, 1-17.

Crossrefs

Same as A024482 except for first term.

Cf. A000108, A000712, A026010, A051924, A113682, A209561.

Programs

GAP

Flat(List([1..30], n->Binomial(2*n-3, n-1)+Binomial(2*n-2, n-2))); # Stefano Spezia, Oct 27 2018

Haskell

a097613 n = a209561 (2 * n - 1) n  -- Reinhard Zumkeller, Dec 26 2012

Magma

[((3*n-2)*Catalan(n-1)+0^(n-1))/2: n in [1..40]]; // G. C. Greubel, Apr 04 2024

Maple

Z:=(1-z-sqrt(1-4*z))/sqrt(1-4*z)/2: Zser:=series(Z, z=0, 32): seq (ceil(coeff(Zser, z, n)), n=1..22); # Zerinvary Lajos, Jan 16 2007
a := n -> `if`(n=1, 1, (2-3*n)/(4-8*n)*binomial(2*n, n)):
seq(a(n), n=1..27); # Peter Luschny, Sep 06 2014

Mathematica

a[1]=1; a[n_] := (3n-2)(2n-3)!/(n!(n-2)!); Array[a, 27] (* Jean-François Alcover, Oct 27 2018 *)

PARI

a(n)=binomial(2*n-3,n-1)+binomial(2*n-2,n-2) \\ Charles R Greathouse IV, Aug 05 2013

Sage

@CachedFunction
def A097613(n):
    if n < 3: return n
    return (6*n-4)*(2*n-3)*A097613(n-1)/(n*(3*n-5))
[A097613(n) for n in (1..27)] # Peter Luschny, Sep 06 2014

Formula

G.f.: (x-1)*(1 - 1/sqrt(1-4*x))/2.

a(n) = ceiling(A051924(n)/2). - Zerinvary Lajos, Jan 16 2007

Integral representation as n-th moment of a signed weight function W(x) = W_a(x) + W_c(x), where W_a(x) = Dirac(x)/2 is the discrete (atomic) part, and W_c(x) = (1/(2*Pi))*((x-1))*sqrt(1/(x*(4-x))) is the continuous part of W(x) defined on (0,4): a(n) = Integral_{x=-eps..eps} x^n*W_a(x) + Integral_{x=0..4} x^n*W_c(x) for any eps > 0, n >= 0. W_c(0) = -infinity, W_c(1) = 0 and W_c(4) = infinity. For 0 < x < 1, W_c(x) < 0, and for 1 < x < 4, W_c(x) > 0. - Karol A. Penson, Aug 05 2013

From Peter Luschny, Sep 06 2014: (Start)

a(n) = ((2-3*n)/(4-8*n))*binomial(2*n,n) for n >= 2.

D-finite with recurrence: a(n) = (6*n-4)*(2*n-3)*a(n-1)/(n*(3*n-5)) for n >= 3. (End)

a(n) ~ 3*2^(2*n-3)/sqrt(n*Pi). - Stefano Spezia, May 09 2023

From G. C. Greubel, Apr 04 2024: (Start)

a(n) = (1/2)*( (3*n-2)*A000108(n-1) + [n=1]).

E.g.f.: (1/2)*(-1+x + exp(2*x)*((1-x)*BesselI(0,2*x) + x*BesselI(1,2*x) )). (End)