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A127530 Triangle read by rows: T(n,k) is the number of binary trees with n edges and k jumps (n >= 0, 0 <= k <= max(0,ceiling(n/2)-1) ).

%I A127530 #24 Feb 08 2025 19:39:01
%S A127530 1,2,5,12,2,29,13,70,60,2,169,235,25,408,836,184,2,985,2790,1046,41,
%T A127530 2378,8896,5080,440,2,5741,27410,22164,3410,61,13860,82230,89440,
%U A127530 21580,900,2,33461,241467,340058,118714,9115,85,80782,696732,1233562,588952
%N A127530 Triangle read by rows: T(n,k) is the number of binary trees with n edges and k jumps (n >= 0, 0 <= k <= max(0,ceiling(n/2)-1) ).
%C A127530 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.
%C A127530 The Krandick reference considers the statistic "number of jumps" for full binary trees.
%C A127530 Row 0 has one term, row n (n >= 1) has ceiling(n/2) terms.
%H A127530 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 A127530 G.f.: G = G(t,z) is given by G = 1 + 2zG + z^2*(t*(G-1)+1)*G.
%F A127530 Row sums are the Catalan numbers (A000108(n+1)).
%F A127530 T(n,0) = A000129(n+1) (the Pell numbers).
%F A127530 Sum_{k>=0} k*T(n,k) = binomial(2*n, n-2) - binomial(2n-2,n-2) = A127531(n).
%e A127530 Triangle starts:
%e A127530     1;
%e A127530     2;
%e A127530     5;
%e A127530    12,   2;
%e A127530    29,  13;
%e A127530    70,  60,  2;
%e A127530   169, 235, 25;
%p A127530 G:= (-z^2-2*z+z^2*t+1-sqrt(z^4+4*z^3-2*z^4*t+2*z^2-4*z^3*t-4*z+z^4*t^2-2*z^2*t+1))/2/t/z^2: Gser:=simplify(series(G,z=0,17)): for n from 1 to 14 do P[n]:=sort(coeff(Gser,z,n)) od: 1; for n from 0 to 14 do seq(coeff(P[n],t,j),j=0..ceil(n/2)-1) od; # yields sequence in triangular form
%t A127530 n = 13; g[t_, z_] := (-z^2 - 2z + z^2*t + 1 - Sqrt[z^4 + 4z^3 - 2z^4*t + 2z^2 - 4z^3*t - 4z + z^4*t^2 - 2z^2*t + 1])/2/t/z^2; Flatten[ CoefficientList[#1, t] & /@ CoefficientList[Simplify[Series[g[t, z], {z, 0, n}]], z]] (* _Jean-François Alcover_, Jul 22 2011, after g.f. *)
%Y A127530 Cf. A000108, A000129, A127531.
%K A127530 nonn,tabf
%O A127530 0,2
%A A127530 _Emeric Deutsch_, Jan 18 2007