%I #11 Nov 21 2013 12:48:47
%S 2,4,2,8,12,2,16,48,24,2,32,160,160,40,2,64,480,800,400,60,2,128,1344,
%T 3360,2800,840,84,2,256,3584,12544,15680,7840,1568,112,2,512,9216,
%U 43008,75264,56448,18816,2688,144,2,1024,23040,138240,322560,338688,169344
%N Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.
%C Row sums are the large Schroeder numbers (A006318). Sum(k*T(n,k),k=1..n)=A002003(n). T(n,k)=2*A114656(n,k).
%F T(n, k)=2^(n-k+1)*binomial(n, k)*binomial(n, k-1)/n (1<=k<=n). G.f. G=G(t, z) satisfies G=z(2+G)(t+G).
%e T(3,3)=2 because we have (U)D(U)D(H) and (U)D(U)D(U)D, where U=(1,1), D=(1,-1) and H=(2,0) (the weak ascents are shown between parentheses).
%e Triangle starts:
%e 2;
%e 4,2;
%e 8,12,2;
%e 16,48,24,2;
%e 32,160,160,40,2.
%p T:=(n,k)->2^(n-k+1)*binomial(n,k)*binomial(n,k-1)/n: for n from 1 to 10 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
%t Flatten[Table[2^(n-k+1) Binomial[n,k] Binomial[n,k-1]/n,{n,10}, {k,n}]] (* _Harvey P. Dale_, Oct 01 2011 *)
%Y Cf. A006318, A002003, A114656.
%K nonn,tabl
%O 1,1
%A _Emeric Deutsch_, Dec 23 2005
%E Keyword tabf changed to tabl by _Michel Marcus_, Apr 09 2013