OFFSET
1,2
COMMENTS
a(n) is the number of Dyck (n-1)-paths that contain no DUDU's and no UUDD's (n>=3). For example, a(5)=3 counts UUUDUDDD, UDUUDUDD, UUDUDDUD. - David Callan, Sep 25 2006
Apart from the first three terms, the total number of bargraphs of semiperimeter n with no levels for n>=2. The sequence begins: 1, 3, 6, 14, 33, ... - Arnold Knopfmacher, Nov 01 2016
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..2192
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.
Aubrey Blecher, Charlotte Brennan, Arnold Knopfmacher, Levels in bargraphs, Ars Mathematica Contemporanea, 9 (2015), 287-300.
FORMULA
G.f.: (1 + x + 2*x^2 - sqrt(1 - 2*x - 3*x^2 + 4*x^4))/2. - Michael Somos, Jun 08 2000
D-finite with recurrence n*(n+1)*a(n) +n*(n+1)*a(n-1) +(n^2+n+12)*a(n-2) +(-29*n^2+85*n+78)*a(n-3) +2*(-13*n^2+56*n+33)*a(n-4) +12*(n+1)*(n-7)*a(n-5) +8*(5*n+3)*(n-8)*a(n-6)=0. - R. J. Mathar, Feb 25 2015
MATHEMATICA
CoefficientList[Series[(1 + x + 2 x^2 - Sqrt[1 - 2 x - 3 x^2 + 4 x^4]) / 2 / x, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 15 2015 *)
PROG
(PARI) a(n)=polcoeff((x+2*x^2-sqrt(1-2*x-3*x^2+4*x^4+x*O(x^n)))/2, n)
(Magma) a:=[1, 2, 1]; for n in [4..35] do Append(~a, &+[a[k]*a[n-k]:k in [1..n-3]] ); end for; a; // Marius A. Burtea, Jan 02 2020
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
STATUS
approved