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A110220 Triangle read by rows: T(n,k) (0 <= k <= floor(n/2)) is the number of lattice paths from (0,0) to (2n,0) consisting of steps U=(1,1), D=(1,-1), H=(2,0), never going below the x-axis (i.e., Schroeder paths) and having k UH's.

%I A110220 #10 Jul 23 2017 02:41:33
%S A110220 1,2,5,1,15,7,51,37,2,188,181,25,731,866,204,5,2950,4124,1393,91,
%T A110220 12235,19657,8672,1008,14,51822,93937,51147,8856,336,223191,450220,
%U A110220 291470,68085,4710,42,974427,2163910,1622665,480535,50655,1254,4302645
%N A110220 Triangle read by rows: T(n,k) (0 <= k <= floor(n/2)) is the number of lattice paths from (0,0) to (2n,0) consisting of steps U=(1,1), D=(1,-1), H=(2,0), never going below the x-axis (i.e., Schroeder paths) and having k UH's.
%C A110220 Row n has 1 + floor(n/2) terms.
%C A110220 Row sums yield the large Schroeder numbers (A006318).
%F A110220 T(n,0) = A007317(n+1).
%F A110220 Sum_{k=0..floor(n/2)} k*T(n,k) = A026002(n-1) for n >= 2.
%F A110220 T(2n,n) = Cat(n) (the n-th Catalan number, A000108).
%F A110220 G.f.: (1 - z - sqrt(1 - 6z + 5z^2 - 4tz^2))/(2z(1 - z + tz)).
%F A110220 T(n,k) = binomial(n+1,k)*Sum_{j=0..n-2k} (binomial(n+1-k, k+j+1)*binomial(2j+2k, j))/(n+1). - _Emeric Deutsch_, Feb 28 2007
%e A110220 T(3,1)=7 because we have HUHD, UHDH, UHDUD, UHHD, UHUDD, UUHDD and UDUHD.
%e A110220 Triangle starts:
%e A110220     1;
%e A110220     2;
%e A110220     5,   1;
%e A110220    15,   7;
%e A110220    51,  37,   2;
%e A110220   188, 181,  25;
%p A110220 G:=(1-z-sqrt(1-6*z+5*z^2-4*z^2*t))/2/z/(1-z+t*z): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 12 do seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)) od; # yields sequence in triangular form
%Y A110220 Cf. A000108, A006318, A007317, A026002.
%K A110220 nonn,tabf
%O A110220 0,2
%A A110220 _Emeric Deutsch_, Jul 16 2005