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A097613 a(n) = binomial(2n-3,n-1) + binomial(2n-2,n-2). 16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) = 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) = 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

LINKS

Table of n, a(n) for n=1..27.

G.-S. Cheon, H. Kim, L. W. Shapiro, Mutation effects in ordered trees, arXiv preprint arXiv:1410.1249 [math.CO], 2014.

Milan Janjic, Two Enumerative Functions

FORMULA

G.f. (x-1)(1-1/sqrt[1-4x])/2.

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

Conjecture: n*a(n) +(-5*n+4)*a(n-1) +2*(2*n-5)*a(n-2)=0. - R. J. Mathar, Nov 26 2012

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) = int(x^n*W_a(x),x=-eps..eps) + int(x^n*W_c(x), x = 0..4) 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

a(n) = ((2-3*n)/(4-8*n))*binomial(2*n,n) for n >= 2. - Peter Luschny, Sep 06 2014

Recurrence: a(n) = (6*n-4)*(2*n-3)*a(n-1)/(n*(3*n-5)) for n >= 3. - Peter Luschny, Sep 06 2014

EXAMPLE

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

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-Fran├žois Alcover, Oct 27 2018 *)

PROG

(Haskell)

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

(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

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

CROSSREFS

Same as A024482 except for first term.

Cf. A000712, A026010, A051924.

Sequence in context: A052936 A108152 A024482 * A074605 A292613 A108081

Adjacent sequences:  A097610 A097611 A097612 * A097614 A097615 A097616

KEYWORD

nonn

AUTHOR

David Callan, Sep 20 2004

STATUS

approved

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Last modified December 11 17:01 EST 2018. Contains 318049 sequences. (Running on oeis4.)