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Search: id:A126216
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| A126216 |
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Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n>=1; 0<=k<=n-1). A Schroeder path of semilength n is a lattice path in the first quadrant, from the origin to the point (2n,0) and consisting of steps U=(1,1), D=(1,-1) and H=(2,0). |
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+0
7
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| 1, 2, 1, 5, 5, 1, 14, 21, 9, 1, 42, 84, 56, 14, 1, 132, 330, 300, 120, 20, 1, 429, 1287, 1485, 825, 225, 27, 1, 1430, 5005, 7007, 5005, 1925, 385, 35, 1, 4862, 19448, 32032, 28028, 14014, 4004, 616, 44, 1, 16796, 75582, 143208, 148512, 91728, 34398, 7644, 936, 54
(list; table; graph; listen)
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OFFSET
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1,2
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COMMENT
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Also number of Schroeder paths of semilength n containing exactly k doublerises but no (2,0) steps at level 0 (n>=1; 0<=k<=n-1). Also number of doublerise-bicolored Dyck paths (doublerises come in two colors; called also marked Dyck paths) of semilength n and having k doublerises of a given color (n>=1; 0<=k<=n-1). Also number of 12312- and 121323-avoiding matchings on [2n] with exactly k crossings.
Essentially the triangle given by [1,1,1,1,1,1,1,1,...] DELTA [0,1,0,1,0,1,0,1,0,1,...] where DELTA is the operator defined in A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 20 2007
Mirror image of triangle A033282 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 20 2007
For relation to Lagrange inversion, or series reversion and the geometry of associahedra, or Stasheff polytopes, (and other combinatorial objects) see A133437. [From Tom Copeland (tcjpn(AT)msn.com), Sep 29 2008]
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REFERENCES
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W. Y. C. Chen, T. Mansour and S. H. F. Yan, Matchings avoiding partial patterns, The Electronic Journal of Combinatorics 13, 2006, #R112, Theorem 3.3.
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LINKS
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D. Callan, Polygon Dissections and Marked Dyck Paths
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FORMULA
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T(n,k)=C(n,k)C(2n-k,n+1)/n (0<=k<=n-1). G.f.=G(t,z)=[1-2z-tz-sqrt(1-4z-2tz+t^2*z^2)]/[2(1+t)z].
Equals N * P, where N = the Narayana triangle (A001263) and P = Pascal's triangle, as infinite lower triangular matrices. A126182 = P * N. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 30 2007
G.f.: 1/(1-x-(x+xy)/(1-xy/(1-(x+xy)/(1-xy/(1-(x+xy)/(1-xy/(1-.... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Feb 06 2009]
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EXAMPLE
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T(3,1)=5 because we have HUUDD, UDHH, UUUDDD, UHUDD and UUDHD.
Triangle starts:
1;
2,1;
5,5,1;
14,21,9,1;
42,84,56,14,1;
Triangle [1,1,1,1,1,1,1,...] DELTA [0,1,0,1,0,1,0,1,...] begins:
1;
1, 0;
2, 1, 0;
5, 5, 1, 0;
14, 21, 9, 1, 0;
42, 84, 56, 14, 1, 0 ;...
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MAPLE
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T:=(n, k)->binomial(n, k)*binomial(2*n-k, n+1)/n: for n from 1 to 11 do seq(T(n, k), k=0..n-1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000108, A002054, A002055, A002056, A007160, A033280, A033281.
Cf. A126182.
Sequence in context: A126124 A060920 A107842 this_sequence A124733 A137597 A059340
Adjacent sequences: A126213 A126214 A126215 this_sequence A126217 A126218 A126219
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 20 2006
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