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Search: id:A101281
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| A101281 |
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Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k low humps. |
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1
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| 1, 1, 1, 2, 3, 1, 8, 8, 5, 1, 36, 28, 18, 7, 1, 164, 120, 68, 32, 9, 1, 764, 552, 292, 136, 50, 11, 1, 3652, 2616, 1356, 608, 240, 72, 13, 1, 17852, 12680, 6532, 2880, 1140, 388, 98, 15, 1, 88868, 62664, 32156, 14128, 5572, 1976, 588, 128, 17, 1, 449004, 314744
(list; table; graph; listen)
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OFFSET
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0,4
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COMMENT
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A Schroeder path of length 2n is a lattice path starting from (0,0), ending at (2n,0), consisting only of steps U=(1,1) (up steps), D=(1,-1) (down steps) and H=(2,0) (level steps) and never going below the x-axis. A hump is an up step U followed by 0 or more level steps H followed by a down step D. A low hump is a hump that starts at height zero. Schroeder paths are counted by the large Schroeder numbers (A006318). Row sums are the large Schroeder numbers (A006318). Column 0 yields A089387.
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FORMULA
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G.f.=G(t, z)=(1-z)R/[1-z+(1-t)zR], where R=[1-z-sqrt(1-6z+z^2)]/(2z) is the g.f. of the large Schroeder numbers (A006318).
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EXAMPLE
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T(3,2)=5 because we have (UD)(UHD), (UHD)(UD), H(UD)(UD), (UD)H(UD) and
(UD)(UD)H, the low humps being shown between parentheses.
Triangle begins:
1;
1,1;
2,3,1;
8,8,5,1;
36,28,18,7,1;
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MAPLE
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G:=(-1+z)*(-1+z+sqrt(1-6*z+z^2))/z/(3-3*z-sqrt(1-6*z+z^2)-t+t*z+t*sqrt(1-6*z+z^2\ )):Gser:=simplify(series(G, z=0, 12)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gser, z^n) od: seq(seq(coeff(t*P[n], t^k), k=1..n+1), n=0..10);
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CROSSREFS
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Cf. A006318, A089387.
Sequence in context: A011152 A078298 A096063 this_sequence A106033 A121634 A006015
Adjacent sequences: A101278 A101279 A101280 this_sequence A101282 A101283 A101284
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch and Ira Gessel, Dec 20 2004
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