cp's OEIS Frontend

A126219 Triangle read by rows: T(n,k) is the number of binary trees (i.e., a rooted tree where each vertex has either 0, 1, or 2 children; and, when only one child is present, it is either a right child or a left child) with n edges and k pairs of adjacent vertices of outdegree 2.

%I A126219 #8 Jul 22 2017 08:36:09
%S A126219 1,2,5,14,40,2,116,16,344,80,5,1040,340,50,3188,1360,300,14,9880,5264,
%T A126219 1484,168,30912,19880,6776,1176,42,97520,73728,29568,6608,588,309856,
%U A126219 269952,124656,33600,4704,132,990656,979264,511584,161280,29544,2112
%N A126219 Triangle read by rows: T(n,k) is the number of binary trees (i.e., a rooted tree where each vertex has either 0, 1, or 2 children; and, when only one child is present, it is either a right child or a left child) with n edges and k pairs of adjacent vertices of outdegree 2.
%C A126219 Row n has floor(n/2) terms (n >= 2).
%C A126219 Row sums are the Catalan numbers (A000108).
%F A126219 T(n,0) = A126220(n).
%F A126219 Sum_{k=0..floor(n/2)-1} k*T(n,k) = 2*binomial(2n-2,n-4) = 2*A002696(n-1) (n >= 4).
%F A126219 G.f.: G = G(t,z) satisfies G = 1 + 2zG + z^2*(1 + 2zG + t(G - 2zG - 1))^2 (see the Maple program for the explicit expression).
%e A126219 Triangle starts:
%e A126219      1;
%e A126219      2;
%e A126219      5;
%e A126219     14;
%e A126219     40,   2;
%e A126219    116,  16;
%e A126219    344,  80,   5;
%e A126219   1040, 340,  50;
%p A126219 G:=1/2*(1-4*z^3*t^2-4*z^3-2*z^2*t+8*z^3*t-2*z+2*z^2*t^2-sqrt(1-8*z^3+4*z^2-4*z^2*t-4*z+8*z^3*t))/z^2/(2*z*t-t-2*z)^2: Gser:=simplify(series(G,z=0,18)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) od: 1;2; for n from 2 to 14 do seq(coeff(P[n],t,j),j=0..floor(n/2)-1) od; # yields sequence in triangular form
%Y A126219 Cf. A000108, A002696, A126220.
%K A126219 nonn,tabf
%O A126219 0,2
%A A126219 _Emeric Deutsch_, Dec 25 2006, Aug 17 2008