A171848 - OEIS

%I #13 Jul 20 2017 01:55:26

%S 1,1,1,1,1,1,2,1,1,3,3,1,1,4,6,4,2,1,5,10,10,7,3,1,1,6,15,20,18,12,7,

%T 2,1,1,7,21,35,39,33,24,14,7,3,1,1,8,28,56,75,76,65,48,32,18,10,4,2,1,

%U 9,36,84,132,156,153,131,102,72,47,28,16,7,3,1,1,10,45,120,217,294,326

%N Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n for which the area below the level steps (i.e., the sum of the altitudes of the level steps) is k (n>=0, k>=0).

%C The considered statistic (area below level steps) in RNA secondary structure terminology is called concentration (see the Willenbring reference, p. 1610).

%C Row n has 1 + Sum_{k=0..n+1} floor(k/4) entries.

%C Sum of entries in row n = A004148(n) (the secondary structure numbers).

%C Sum_{k>=0} k*T(n,k) = A171849(n).

%H R. Willenbring, <a href="https://doi.org/10.1016/j.dam.2008.10.002">RNA structure, permutations and statistics</a>, Discrete Appl. Math., 157, 2009, 1607-1614.

%F The trivariate g.f. G=G(t,u,z), where z marks length, t marks area below the level steps, and u marks number of level steps, satisfies G(t,u,z) = 1 + uzG(t,u,z) + z^2*(G(t,tu,z) - 1)G(t,u,z).

%e T(4,2)=1 because we have UHHD, where U=(1,1), H=(1,0), D=(1,-1).

%e Triangle starts:

%e   1;

%e   1;

%e   1;

%e   1,  1;

%e   1,  2,  1;

%e   1,  3,  3,  1;

%e   1,  4,  6,  4,  2;

%e   1,  5, 10, 10,  7,  3,  1;

%p g[0] := 1/(1-u*z+z^2-z^2*g[1]): for n to 15 do g[n] := subs({u = t*u, g[n] = g[n+1]}, g[n-1]) end do: G := subs({u = 1, g[16] = 0}, g[0]): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. sum(floor((1/4)*j), j = 0 .. n+1)) end do; # yields sequence in triangular form

%Y Cf. A004148, A171849.

%K nonn,tabf

%O 0,7

%A _Emeric Deutsch_, Feb 08 2010