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Search: id:A100754
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| A100754 |
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Triangle read by rows: T(n,k)=number of hill-free Dyck paths (i.e. no peaks at height 1) of semilength n and having k peaks. |
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+0
3
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| 1, 1, 1, 1, 4, 1, 1, 8, 8, 1, 1, 13, 29, 13, 1, 1, 19, 73, 73, 19, 1, 1, 26, 151, 266, 151, 26, 1, 1, 34, 276, 749, 749, 276, 34, 1, 1, 43, 463, 1781, 2762, 1781, 463, 43, 1, 1, 53, 729, 3758, 8321, 8321, 3758, 729, 53, 1, 1, 64, 1093, 7253, 21659, 31004, 21659, 7253, 1093, 64, 1
(list; table; graph; listen)
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OFFSET
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2,5
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COMMENT
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Row n has n-1 terms. Row sums yield the Fine numbers (A000957).
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REFERENCES
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E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
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FORMULA
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T(n, k)=sum([j/(n-j)]binomial(n-j, k-j)binomial(n-j, k), j=0..min(k, n-k)) (n>=2). G.f.=tzr/(1-tzr), where r=r(t, z) is the Narayana function defined by r=z(1+r)(1+tr).
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EXAMPLE
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T(4,2)=4 because we have UU*DDUU*DD, UU*DUU*DDD, UUU*DDU*DD, and UUU*DU*DDD, where U=(1,1), D=(1,-1) and * indicates the peaks.
Triangle starts:
1;
1,1;
1,4,1;
1,8,8,1;
1,13,29,13,1;
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MAPLE
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T:=(n, k)->sum((j/(n-j))*binomial(n-j, k-j)*binomial(n-j, k), j=0..min(k, n-k)): for n from 2 to 13 do seq(T(n, k), k=1..n-1) od; # yields the sequence in triangular form
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CROSSREFS
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Cf. A000957.
Sequence in context: A119673 A051455 A132789 this_sequence A055107 A128137 A136100
Adjacent sequences: A100751 A100752 A100753 this_sequence A100755 A100756 A100757
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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), Jan 14 2005
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