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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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4
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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
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OFFSET
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0,5
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COMMENTS
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Column 0 is A004148 (RNA secondary structure numbers).
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REFERENCES
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Cameron, Naiomi, and Everett Sullivan. "Peakless Motzkin paths with marked level steps at fixed height." Discrete Mathematics 344.1 (2021): 112154.
He, Tian-Xiao. "A-sequences, Z-sequence, and B-sequences of Riordan matrices." Discrete Mathematics 343.3 (2020): 111718.
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LINKS
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A. Panayotopoulos and P. Vlamos, Cutting Degree of Meanders, Artificial Intelligence Applications and Innovations, IFIP Advances in Information and Communication Technology, Volume 382, 2012, pp 480-489; DOI 10.1007/978-3-642-33412-2_49. - From N. J. A. Sloane, Dec 29 2012
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FORMULA
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T(n,k) = (k+1)*Sum_{j=ceiling((n-k+1)/2)..n-k} (C(j,n-k-j)*C(j+k,n+1-j)/j) 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.
T(n,k) = T(n-1,k-1) + Sum_{j>=0} T(n-1-j,k+j), T(0,0) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 26 2014
Sum_{j=0..n-1} cos(2*Pi*k/3 + Pi/6)*T(n,k) = cos(Pi*n/2)*sqrt(3)/2 - cos(2*Pi*n/3 + Pi/6). - Leonid Bedratyuk, Dec 06 2017
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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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MATHEMATICA
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T[n_, k_] := T[n, k] = If[k==n, 1, (k+1)*Sum[Binomial[j, n-k-j]*Binomial[j +k, n+1-j]/j, {j, Ceiling[(n-k+1)/2], n-k}]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 22 2017, translated from Maple *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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