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%I #14 Mar 03 2020 12:31:11
%S 1,0,1,1,0,1,0,3,0,1,1,0,6,0,1,0,6,0,10,0,1,1,0,20,0,15,0,1,0,10,0,50,
%T 0,21,0,1,1,0,50,0,105,0,28,0,1,0,15,0,175,0,196,0,36,0,1,1,0,105,0,
%U 490,0,336,0,45,0,1,0,21,0,490,0,1176,0,540,0,55,0,1,1,0,196,0,1764,0,2520,0
%N Triangle read by rows: T(n,k)(1<=k<=n) is the number of peakless Motzkin paths of length n having k (1,0) steps (can be easily translated into RNA secondary structure terminology).
%C Row sums yield A004148. sum(k*T(n,k),k=1..n)=A110236(n).
%H W. R. Schmitt and M. S. Waterman, <a href="http://dx.doi.org/10.1016/0166-218X(92)00038-N">Linear trees and RNA secondary structure</a>, Discrete Appl. Math., 51, 317-323, 1994.
%H P. R. Stein and M. S. Waterman, <a href="http://dx.doi.org/10.1016/0012-365X(79)90033-5">On some new sequences generalizing the Catalan and Motzkin numbers</a>, Discrete Math., 26 (1978), 261-272.
%H M. Vauchassade de Chaumont and G. Viennot, <a href="https://www.emis.de/journals/SLC/opapers/s08viennot.html">Polynômes orthogonaux et problèmes d'énumération en biologie moléculaire</a>, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.
%F T(n, k) = [2/(n+k)]binomial((n+k)/2, k)*binomial((n+k)/2, k-1).
%F G.f.: g=g(t, z) satisfies g=1+tzg+z^2*g(g-1).
%F G.f.: A(x,y) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2 * y^k * x^(n-k)] * x^n/n ). - _Paul D. Hanna_, Oct 21 2012
%e T(5,3)=6 because we have UHDHH, UHHDH, UHHHD, HUHDH, HUHHD and HHUHD, where U=(1,1), D=(1,-1), H=(1,0).
%e Triangle starts:
%e 1;
%e 0, 1;
%e 1, 0, 1;
%e 0, 3, 0, 1;
%e 1, 0, 6, 0, 1;
%e 0, 6, 0, 10, 0, 1;
%e 1, 0, 20, 0, 15, 0, 1;
%e 0, 10, 0, 50, 0, 21, 0, 1;
%e 1, 0, 50, 0, 105, 0, 28, 0, 1;
%e 0, 15, 0, 175, 0, 196, 0, 36, 0, 1;
%e 1, 0, 105, 0, 490, 0, 336, 0, 45, 0, 1; ...
%p T:=proc(n,k) if n+k mod 2 = 0 then 2*binomial((n+k)/2,k)*binomial((n+k)/2,k-1)/(n+k) else 0 fi end: for n from 1 to 14 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
%o (PARI) T(n,k)=polcoeff(polcoeff(exp(sum(m=1,n,sum(j=0,m,binomial(m,j)^2*y^j*x^(m-j))*x^m/m)+O(x^(n+1))),n,x),k,y)
%o for(n=0,10,for(k=0,n,print1(T(n,k),", "));print()) \\ _Paul D. Hanna_, Oct 21 2012
%Y Cf. A004148, A110236.
%K nonn,tabl
%O 1,8
%A _Emeric Deutsch_, Jul 17 2005
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