A025243 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4.

1, 2, 1, 1, 3, 6, 14, 33, 79, 194, 482, 1214, 3090, 7936, 20544, 53545, 140399, 370098, 980226, 2607242, 6961462, 18652112, 50133616, 135140598, 365254226, 989614976, 2687312752, 7312725944, 19938170096, 54460115308, 149007155356, 408341969073

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

Conjecture: 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

Sequence in context: A060517 A163181 A074662 * A352765 A361616 A341014

Adjacent sequences: A025240 A025241 A025242 * A025244 A025245 A025246

Keyword

nonn

Author

Clark Kimberling

Status

approved