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

%I A097613 #76 Feb 19 2025 12:12:27
%S A097613 1,2,7,25,91,336,1254,4719,17875,68068,260338,999362,3848222,14858000,
%T A097613 57500460,222981435,866262915,3370764540,13135064250,51250632510,
%U A097613 200205672810,782920544640,3064665881940,12007086477750,47081501377326,184753963255176,725510446350004
%N A097613 a(n) = binomial(2n-3,n-1) + binomial(2n-2,n-2).
%C A097613 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.
%C A097613 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
%C A097613 Binomial transform of A113682. - _Paul Barry_, Aug 21 2007
%C A097613 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
%C A097613 Equals the Catalan sequence, A000108, convolved with A051924 prefaced with a 1: (1, 1, 4, 14, 50, ...). - _Gary W. Adamson_, May 15 2009
%C A097613 Central terms of triangle A209561. - _Reinhard Zumkeller_, Dec 26 2012
%C A097613 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
%H A097613 Michael De Vlieger, <a href="/A097613/b097613.txt">Table of n, a(n) for n = 1..1664</a>
%H A097613 Paul Barry, <a href="https://doi.org/10.3390/math13020242">Extensions of Riordan Arrays and Their Applications</a>, Mathematics (2025) Vol. 13, No. 2, 242. See p. 16.
%H A097613 Gi-Sang Cheon, Hana Kim, and Louis W. Shapiro, <a href="http://arxiv.org/abs/1410.1249">Mutation effects in ordered trees</a>, arXiv preprint arXiv:1410.1249 [math.CO], 2014.
%H A097613 Milan Janjic, <a href="https://old.pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>
%H A097613 Toufik Mansour and I. L. Ramirez, <a href="https://ajc.maths.uq.edu.au/pdf/81/ajc_v81_p447.pdf">Enumerations of polyominoes determined by Fuss-Catalan words</a>, Australas. J. Combin. 81 (3) (2021) 447-457, table 2.
%H A097613 Lin Yang and Shengliang Yang, <a href="https://doi.org/10.4208/jms.v56n1.23.01">Protected Branches in Ordered Trees</a>, J. Math. Study (2023) Vol. 56, No. 1, 1-17.
%F A097613 G.f.: (x-1)*(1 - 1/sqrt(1-4*x))/2.
%F A097613 a(n) = ceiling(A051924(n)/2). - _Zerinvary Lajos_, Jan 16 2007
%F A097613 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
%F A097613 From _Peter Luschny_, Sep 06 2014: (Start)
%F A097613 a(n) = ((2-3*n)/(4-8*n))*binomial(2*n,n) for n >= 2.
%F A097613 D-finite with recurrence: a(n) = (6*n-4)*(2*n-3)*a(n-1)/(n*(3*n-5)) for n >= 3. (End)
%F A097613 a(n) ~ 3*2^(2*n-3)/sqrt(n*Pi). - _Stefano Spezia_, May 09 2023
%F A097613 From _G. C. Greubel_, Apr 04 2024: (Start)
%F A097613 a(n) = (1/2)*( (3*n-2)*A000108(n-1) + [n=1]).
%F A097613 E.g.f.: (1/2)*(-1+x + exp(2*x)*((1-x)*BesselI(0,2*x) + x*BesselI(1,2*x) )). (End)
%e A097613 a(2) = 2 because UUDDUD and UDUUDD each have maximum pyramid size = 2.
%p A097613 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
%p A097613 a := n -> `if`(n=1, 1, (2-3*n)/(4-8*n)*binomial(2*n, n)):
%p A097613 seq(a(n), n=1..27); # _Peter Luschny_, Sep 06 2014
%t A097613 a[1]=1; a[n_] := (3n-2)(2n-3)!/(n!(n-2)!); Array[a, 27] (* _Jean-François Alcover_, Oct 27 2018 *)
%o A097613 (Haskell)
%o A097613 a097613 n = a209561 (2 * n - 1) n  -- _Reinhard Zumkeller_, Dec 26 2012
%o A097613 (PARI) a(n)=binomial(2*n-3,n-1)+binomial(2*n-2,n-2) \\ _Charles R Greathouse IV_, Aug 05 2013
%o A097613 (Sage)
%o A097613 @CachedFunction
%o A097613 def A097613(n):
%o A097613     if n < 3: return n
%o A097613     return (6*n-4)*(2*n-3)*A097613(n-1)/(n*(3*n-5))
%o A097613 [A097613(n) for n in (1..27)] # _Peter Luschny_, Sep 06 2014
%o A097613 (GAP) Flat(List([1..30], n->Binomial(2*n-3, n-1)+Binomial(2*n-2, n-2))); # _Stefano Spezia_, Oct 27 2018
%o A097613 (Magma) [((3*n-2)*Catalan(n-1)+0^(n-1))/2: n in [1..40]]; // _G. C. Greubel_, Apr 04 2024
%Y A097613 Same as A024482 except for first term.
%Y A097613 Cf. A000108, A000712, A026010, A051924, A113682, A209561.
%K A097613 nonn
%O A097613 1,2
%A A097613 _David Callan_, Sep 20 2004