A127533 - OEIS

%I #17 Sep 08 2022 08:45:29

%S 0,0,0,2,17,100,506,2366,10556,45696,193800,810084,3350479,13748020,

%T 56071470,227613750,920540040,3711935040,14932102320,59951235420,

%U 240316859250,962056169256,3847193657076,15370712686252,61364157982952

%N Sum of jump-lengths of 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; the positive difference of the levels is called the jump distance; the sum of the jump distances in a given binary tree is called the jump-length.

%C The Krandick reference is about jumps and jump-length in full binary trees.

%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.

%F G.f.: z^3*C^6*(C+1)/sqrt(1-4z), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.

%F a(n) = binomial(2*n+1,n-3) + binomial(2*n,n-3).

%F a(n) = Sum_{k>=0} A127532(n,k).

%F a(n) ~ n -> 4^n*(3-275/(8*n)+29475/(128*n^2)-1268225/(1024*n^3)+195652737/ (32768*n^4))/sqrt(n*Pi). - _Peter Luschny_, Dec 19 2015

%F D-finite with recurrence -(n-3)*(3*n+2)*(n+4)*a(n) +2*n*(3*n+5)*(2*n-1)*a(n-1)=0. - _R. J. Mathar_, Jul 26 2022

%p seq(binomial(2*n+1,n-3)+binomial(2*n,n-3),n=0..28);

%t Table[Binomial[2 n + 1, n - 3] + Binomial[2 n, n - 3], {n, 0, 24}] (* _Michael De Vlieger_, Dec 19 2015 *)

%o (Magma) [Binomial(2*n+1,n-3) + Binomial(2*n,n-3): n in [0..30]]; // _Vincenzo Librandi_, Dec 20 2015

%Y Cf. A127532, A127529.

%K nonn

%O 0,4

%A _Emeric Deutsch_, Jan 18 2007