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A110235
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).
1
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, 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, 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
OFFSET
1,8
COMMENTS
Row sums yield A004148. sum(k*T(n,k),k=1..n)=A110236(n).
LINKS
W. R. Schmitt and M. S. Waterman, Linear trees and RNA secondary structure, Discrete Appl. Math., 51, 317-323, 1994.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1978), 261-272.
M. Vauchassade de Chaumont and G. Viennot, Polynômes orthogonaux et problèmes d'énumération en biologie moléculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.
FORMULA
T(n, k) = [2/(n+k)]binomial((n+k)/2, k)*binomial((n+k)/2, k-1).
G.f.: g=g(t, z) satisfies g=1+tzg+z^2*g(g-1).
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
EXAMPLE
T(5,3)=6 because we have UHDHH, UHHDH, UHHHD, HUHDH, HUHHD and HHUHD, where U=(1,1), D=(1,-1), H=(1,0).
Triangle starts:
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, 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, 490, 0, 336, 0, 45, 0, 1; ...
MAPLE
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
PROG
(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)
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print()) \\ Paul D. Hanna, Oct 21 2012
CROSSREFS
Sequence in context: A229995 A119467 A166353 * A036856 A036855 A147985
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Jul 17 2005
STATUS
approved