A114655 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.
2, 4, 2, 8, 12, 2, 16, 48, 24, 2, 32, 160, 160, 40, 2, 64, 480, 800, 400, 60, 2, 128, 1344, 3360, 2800, 840, 84, 2, 256, 3584, 12544, 15680, 7840, 1568, 112, 2, 512, 9216, 43008, 75264, 56448, 18816, 2688, 144, 2, 1024, 23040, 138240, 322560, 338688, 169344
Offset: 1
Examples
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). Triangle starts: 2; 4,2; 8,12,2; 16,48,24,2; 32,160,160,40,2.
Programs
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Maple
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
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Mathematica
Flatten[Table[2^(n-k+1) Binomial[n,k] Binomial[n,k-1]/n,{n,10}, {k,n}]] (* Harvey P. Dale, Oct 01 2011 *)
Formula
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).
Extensions
Keyword tabf changed to tabl by Michel Marcus, Apr 09 2013
Comments