OFFSET
1,2
COMMENTS
Antidiagonal sums of A132812. - Philippe Deléham, Jun 08 2013
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Jean-Luc Baril, Sergey Kirgizov, Rémi Maréchal, and Vincent Vajnovszki, Grand Dyck paths with air pockets, arXiv:2211.04914 [math.CO], 2022.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Peter McCalla and Asamoah Nkwanta, Catalan and Motzkin Integral Representations, arXiv:1901.07092 [math.NT], 2019.
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, Polynomes orthogonaux et problèmes d'énumeration en biologie moléculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.
FORMULA
G.f.: (1-z-z^2)/(2*z^2*sqrt(1-2*z-z^2-2*z^3+z^4))-1/(2*z^2).
a(n) = Sum_{k=1..n} k*A110319(n,k).
a(n) ~ sqrt(4 + 9/sqrt(5)) * (3+sqrt(5))^n / (sqrt(Pi*n) * 2^(n+1)). - Vaclav Kotesovec, Sep 25 2016, equivalently, a(n) ~ phi^(2*n + 3) / (2 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 06 2021
D-finite with recurrence (n+2)*a(n) +3*(-n-1)*a(n-1) +(n-7)*a(n-3) +2*(2*n-3)*a(n-4) +(n-5)*a(n-5) +(-n+4)*a(n-6)=0. - R. J. Mathar, Feb 21 2020
EXAMPLE
a(4)=13 because the 4 (=A004148(4)) RNA secondary structures of size 4, namely 1/2/3/4, 13/2/4, 14/2/3 and 1/24/3, have altogether 4+3+3+3=13 blocks.
MAPLE
G:=1/2*(1-z-z^2)/z^2/(1-2*z-z^2-2*z^3+z^4)^(1/2)-1/2*1/(z^2): Gser:=series(G, z=0, 37): seq(coeff(Gser, z^n), n=1..33);
MATHEMATICA
Table[Sum[Binomial[n-j+1, j]Binomial[n-j+1, j-1], {j, 0, n}], {n, 1, 25}] (* Benedict W. J. Irwin, Sep 24 2016 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jul 19 2005
STATUS
approved