Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #41 Dec 06 2022 17:14:39
%S 0,0,2,13,64,285,1210,5005,20384,82212,329460,1314610,5230016,
%T 20764055,82317690,326012925,1290244800,5103910680,20183646780,
%U 79802261190,315492902400,1247247742650,4930910180196,19495286167698,77085553829824
%N Number of jumps in all binary trees with n edges.
%C In the preorder traversal of a binary tree, any transition from a node at a deeper level to a node on a strictly higher level is called a jump.
%H G. C. Greubel, <a href="/A127531/b127531.txt">Table of n, a(n) for n = 1..1000</a>
%H David Anderson, E. S. Egge, M. Riehl, L. Ryan, R. Steinke, Y. Vaughan, <a href="http://arxiv.org/abs/1605.06825">Pattern Avoiding Linear Extensions of Rectangular Posets</a>, arXiv:1605.06825 [math.CO], 2016.
%H Colin Defant, <a href="https://arxiv.org/abs/1905.02309">Proofs of Conjectures about Pattern-Avoiding Linear Extensions</a>, arXiv:1905.02309 [math.CO], 2019.
%H W. Krandick, <a href="http://dx.doi.org/10.1016/j.cam.2003.08.018">Trees and jumps and real roots</a>, J. Computational and Applied Math., 162, 2004, 51-55.
%H 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 a(n) = Sum_{k>=0} k*A127530(n,k).
%F a(n) = binomial(2*n, n-2) - binomial(2*n - 2, n-2).
%F From _Peter Luschny_, Dec 19 2015: (Start)
%F a(n) = 4^(n-1)*(n-1)*(3*n^2-5*n-2)*Gamma(n-1/2)/(sqrt(Pi)*Gamma(n+3)).
%F a(n) ~ 4^n*(3-139/(8*n)+8595/(128*n^2)-234745/(1024*n^3)+24282657/(32768*n^4)) /sqrt(n*Pi). (End)
%F D-finite with recurrence -5*(n+2)*(n-3)*a(n) +(19*n^2-26*n-5)*a(n-1) +2*(n-2)*(2*n-5)*a(n-2)=0. - _R. J. Mathar_, Jul 26 2022
%F D-finite with recurrence +(n-3)*(3*n-2)*(n+2)*a(n) -2*(n-1)*(3*n+1)*(2*n-3)*a(n-1)=0. - _R. J. Mathar_, Jul 26 2022
%p seq(binomial(2*n,n-2)-binomial(2*n-2,n-2),n=1..28);
%t Table[Binomial[2n, n-2] - Binomial[2n-2, n-2], {n, 30}] (* or *) Table[4^(n-1)(n-1)(3n^2 -5n-2)Gamma[n-1/2]/(Sqrt[Pi]Gamma[n+3]), {n,30}] (* _Michael De Vlieger_, Dec 19 2015 *)
%o (Magma) [Binomial(2*n, n-2) - Binomial(2*n - 2, n-2): n in [1..30]]; // _Vincenzo Librandi_, Dec 20 2015
%o (PARI) vector(30, n, binomial(2*n,n-2) -binomial(2*n-2,n-2) ) \\ _G. C. Greubel_, Mar 19 2017
%o (Magma) [Binomial(2*n,n-2) -Binomial(2*n-2,n-2): n in [1..30]]; // _G. C. Greubel_, May 08 2019
%o (Sage) [binomial(2*n,n-2) -binomial(2*n-2,n-2) for n in (1..30)] # _G. C. Greubel_, May 08 2019
%o (GAP) List([1..30], n-> Binomial(2*n,n-2) -Binomial(2*n-2,n-2)) # _G. C. Greubel_, May 08 2019
%Y Cf. A127530.
%K nonn
%O 1,3
%A _Emeric Deutsch_, Jan 18 2007