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A114597
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Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n having pyramid weight k.
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0
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1, 1, 1, 1, 3, 2, 1, 5, 9, 3, 1, 7, 21, 23, 5, 1, 9, 38, 74, 56, 8, 1, 11, 60, 170, 237, 130, 13, 1, 13, 87, 325, 674, 706, 293, 21, 1, 15, 119, 553, 1535, 2442, 1994, 645, 34, 1, 17, 156, 868, 3030, 6542, 8259, 5401, 1395, 55, 1, 19, 198, 1284, 5411, 14840, 25738, 26441
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OFFSET
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2,5
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COMMENTS
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A pyramid in a Dyck word (path) is a factor of the form U^h D^h, h being the height of the pyramid. A pyramid in a Dyck word w is maximal if, as a factor in w, it is not immediately preceded by a U and immediately followed by a D. The pyramid weight of a Dyck path (word) is the sum of the heights of its maximal pyramids. Row sums are the Fine numbers (A000957). T(n,n)=fibonacci(n-1) (A000045).
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LINKS
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FORMULA
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G.f.= G-1, where G=G(t, z) satisfies z(1+t-t^2*z-t^3*z^2)G^2-(1+z-2t^2*z^2)G+1-tz=0.
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EXAMPLE
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T(4,3)=3 because we have U(UD)(UUDD)D, U(UUDD)(UD)D and U(UD)(UD)(UD)D, where U=(1,1),D=(1,-1) (the maximal pyramids are shown between parentheses).
Triangle begins:
1;
1,1;
1,3,2;
1,5,9,3;
1,7,21,23,5;
1,9,38,74,56,8;
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MAPLE
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G:=(1+z-2*t^2*z^2-sqrt((1-z)*(1-z-4*t*z+4*t^2*z^2)))/2/z/(1+t-t^2*z-t^3*z^2)-1: Gser:=simplify(series(G, z=0, 15)): for n from 2 to 13 do P[n]:=expand(coeff(Gser, z^n)) od: for n from 2 to 13 do seq(coeff(P[n], t^j), j=2..n) od; # yields sequence in triangular form
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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