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Search: id:A114588
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| A114588 |
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Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k peaks at even levels (0<=k<=n-1; n>=2). A hill in a Dyck path is a peak at level 1. |
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+0
4
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| 0, 1, 1, 0, 1, 1, 3, 1, 1, 3, 6, 6, 2, 1, 7, 17, 18, 11, 3, 1, 17, 48, 58, 40, 18, 4, 1, 43, 134, 186, 150, 76, 27, 5, 1, 110, 380, 590, 540, 325, 130, 38, 6, 1, 286, 1083, 1860, 1915, 1305, 624, 206, 51, 7, 1, 753, 3100, 5844, 6660, 5115, 2772, 1097, 308, 66, 8, 1, 2003
(list; graph; listen)
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OFFSET
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2,7
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COMMENT
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Row n has n terms. Row sums are the Fine numbers (A000957). Column 0 yields A114589. Sum(k*T(n,k),k=0..n-1) yields A114590.
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FORMULA
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G.f.=G-1, where G=G(t, z) satisfies z(2+2z+z^2-tz-tz^2)G^2+(1+2z)(1+z-tz)G+1+z-tz=0.
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EXAMPLE
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T(4,3)=1 because we have U(UD)(UD)(UD)D, where U=(1,1), D=(1,-1) (the peaks at even levels are shown between parentheses).
Triangle begins:
0,1;
1,0,1;
1,3,1,1;
3,6,6,2,1;
7,17,18,11,3,1;
17,48,58,40,18,4,1;
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MAPLE
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G:=(1-t*z+2*z^2+3*z-2*t*z^2-sqrt(1-3*z^2-2*z*t+2*z^2*t+z^2*t^2-2*z))/2/z/(2+2*z-t*z-t*z^2+z^2)-1: Gser:=simplify(series(G, z=0, 15)): for n from 2 to 12 do P[n]:=coeff(Gser, z^n) od: for n from 2 to 12 do seq(coeff(t*P[n], t^j), j=1..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000957, A114586, A114587, A114589, A114590.
Sequence in context: A094644 A113046 A133825 this_sequence A121745 A089312 A125127
Adjacent sequences: A114585 A114586 A114587 this_sequence A114589 A114590 A114591
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 11 2005
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