

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 xaxis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.


3



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
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OFFSET

1,1


COMMENTS

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).


LINKS

Table of n, a(n) for n=1..51.


FORMULA

T(n, k)=2^(nk+1)*binomial(n, k)*binomial(n, k1)/n (1<=k<=n). G.f. G=G(t, z) satisfies G=z(2+G)(t+G).


EXAMPLE

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.


MAPLE

T:=(n, k)>2^(nk+1)*binomial(n, k)*binomial(n, k1)/n: for n from 1 to 10 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form


MATHEMATICA

Flatten[Table[2^(nk+1) Binomial[n, k] Binomial[n, k1]/n, {n, 10}, {k, n}]] (* Harvey P. Dale, Oct 01 2011 *)


CROSSREFS

Cf. A006318, A002003, A114656.
Sequence in context: A308044 A319252 A114593 * A228890 A051288 A120434
Adjacent sequences: A114652 A114653 A114654 * A114656 A114657 A114658


KEYWORD

nonn,tabl


AUTHOR

Emeric Deutsch, Dec 23 2005


EXTENSIONS

Keyword tabf changed to tabl by Michel Marcus, Apr 09 2013


STATUS

approved



