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A097724 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.

%I A097724 #46 Jun 26 2025 07:36:38
%S A097724 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,
%T A097724 62,69,59,40,21,7,1,82,140,160,144,105,62,28,8,1,185,320,375,350,271,
%U A097724 174,91,36,9,1,423,740,885,852,690,474,273,128,45,10,1,978,1728,2102,2077
%N A097724 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.
%C A097724 Column 0 is A004148 (RNA secondary structure numbers).
%C A097724 This triangle appears identical to A191579 (apart from offsets). - _Philippe Deléham_, Jan 26 2014
%C A097724 Conjecture: the row reverse triangle is the triangle of connection constants for expressing the polynomial u(n,x+1) as a linear combination of the polynomials u(k,x), 0 <= k <= n, where u(n,x) = U(n,x/2) with U(n,x) the n-th Chebyshev polynomial of the second kind. An example is given below.  Cf. A205810. - _Peter Bala_, Jun 26 2025
%H A097724 Alois P. Heinz, <a href="/A097724/b097724.txt">Rows n = 0..140, flattened</a>
%H A097724 Naiomi T. Cameron and Asamoah Nkwanta, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Cameron/cameron46.html">On Some (Pseudo) Involutions in the Riordan Group</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.
%H A097724 Naiomi Cameron and Everett Sullivan, <a href="https://doi.org/10.1016/j.disc.2020.112154">Peakless Motzkin paths with marked level steps at fixed height</a>, Discrete Mathematics 344.1 (2021): 112154.
%H A097724 Tian-Xiao He, <a href="https://doi.org/10.1016/j.disc.2019.111718">A-sequences, Z-sequence, and B-sequences of Riordan matrices</a>, Discrete Mathematics 343.3 (2020): 111718.
%H A097724 A. Nkwanta and A. Tefera, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Nkwanta/nkwanta4.html">Curious Relations and Identities Involving the Catalan Generating Function and Numbers</a>, Journal of Integer Sequences, 16 (2013), #13.9.5.
%H A097724 A. Panayotopoulos and P. Vlamos, <a href="http://dx.doi.org/10.1007/978-3-642-33412-2_49">Cutting Degree of Meanders</a>, 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
%F A097724 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.
%F A097724 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.
%F A097724 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
%F A097724 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
%e A097724 Triangle starts:
%e A097724   1;
%e A097724   1, 1;
%e A097724   1, 2, 1;
%e A097724   2, 3, 3, 1;
%e A097724   4, 6, 6, 4, 1;
%e A097724 Row n has n+1 terms.
%e A097724 T(3,2)=3 because we have HUU, UHU and UUH, where U=(1,1) and H=(1,0).
%e A097724 Row 7: let u(n,x) = U(n,x/2). Then u(7,x+1) = u(7,x) + 7*u(6,x) + 21*u(5,x) + 40*u(4,x) + 59*u(3,x) + 69*u(2,x) + 62*u(1,x) + 37. - _Peter Bala_, Jun 26 2025
%p A097724 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
%t A097724 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 *)
%Y A097724 Cf. A004148, A191579, A091964 (row sums), A205810.
%K A097724 nonn,tabl
%O A097724 0,5
%A A097724 _Emeric Deutsch_, Sep 11 2004