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Search: id:A097724
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| A097724 |
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Triangle read by rows: T(n,k) is the number of left factors of Motzkin paths without peaks, having length n and endpoint height k. |
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+0
3
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| 1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 4, 6, 6, 4, 1, 8, 13, 13, 10, 5, 1, 17, 28, 30, 24, 15, 6, 1, 37, 62, 69, 59, 40, 21, 7, 1, 82, 140, 160, 144, 105, 62, 28, 8, 1, 185, 320, 375, 350, 271, 174, 91, 36, 9, 1, 423, 740, 885, 852, 690, 474, 273, 128, 45, 10, 1, 978, 1728, 2102, 2077
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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Column 0 is A004148 (RNA secondary structure numbers).
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REFERENCES
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Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.
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FORMULA
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T(n, k)=(k+1)sum(binomial(j, n-k-j)*binomial(j+k, n+1-j)/j, j=ceil((n-k+1)/2)..n-k) for 0<=k<n; T(n, n)=1. G.f.=G/(1-tzG), where G = [1-z+z^2-sqrt(1-2z-z^2-2z^3+z^4)]/(2z^2) is the g.f. for the sequence A004148.
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EXAMPLE
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Triangle starts:
1;
1,1;
1,2,1;
2,3,3,1;
4,6,6,4,1;
Row n has n+1 terms.
T(3,2)=3 because we have HUU, UHU and UUH, where U=(1,1) and H=(1,0).
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MAPLE
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T:=proc(n, k) if k=n then 1 else (k+1)*sum(binomial(j, n-k-j)*binomial(j+k, n+1-j)/j, j=ceil((n-k+1)/2)..n-k) fi end: seq(seq(T(n, k), k=0..n), n=0..12); T:=proc(n, k) if k=n then 1 else (k+1)*sum(binomial(j, n-k-j)*binomial(j+k, n+1-j)/j, j=ceil((n-k+1)/2)..n-k) fi end: TT:=(n, k)->T(n-1, k-1): matrix(10, 10, TT); # gives the sequence as a matrix
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CROSSREFS
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Cf. A004148.
Sequence in context: A140530 A052250 A099569 this_sequence A091836 A080850 A109449
Adjacent sequences: A097721 A097722 A097723 this_sequence A097725 A097726 A097727
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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), Sep 11 2004
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