A273905 Number of symmetric bargraphs having semiperimeter n (n>=2).

1, 2, 3, 5, 9, 15, 27, 46, 83, 143, 259, 450, 817, 1429, 2599, 4570, 8323, 14698, 26797, 47491, 86659, 154042, 281287, 501283, 915907, 1635835, 2990383, 5351138, 9786369, 17541671, 32092959, 57610988, 105435607, 189521640, 346950321, 624389105

Offset: 2

Author

Emeric Deutsch and Sergi Elizalde, Jun 23 2016

Keywords

• nonn

Examples

			a(4) = 3; indeed, the corresponding compositions are [3],[2,2],[1,1,1].
a(6) = 9; indeed, the corresponding compositions are [5],[4,4],[1,3,1],[2,3,2],[2,1,2],[3,3,3],[2,2,2,2],[1,2,2,1],[1,1,1,1,1].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..2000
• M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.
• Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088, 2016

Programs

Maple

Q := sqrt((1-z^2)*(1-z-z^2-z^3)*(1+z-z^2+z^3)): g := (1/2)*(1+z)*(z^4+2*z^3+2*z^2-1+Q)/(z*(1-z-z^2-z^3)): gser := series(g, z = 0, 42): seq(coeff(gser,z,n), n=2..37);
# second Maple program:
a:= proc(n) option remember; `if`(n<9, [$0..3, 5, 9, 15, 27]
      [n], (2*a(n-1) +(4*n-6)*a(n-2) -(2*n-12)*a(n-4)
      -6*a(n-5) +2*a(n-6) -(n-9)*a(n-8))/ (n+1))
    end:
seq(a(n), n=2..40);  # Alois P. Heinz, Jun 24 2016

Mathematica

a[2]=1; a[3]=2; a[4]=3; a[5]=5; a[6]=9; a[7]=15; a[8]=27; a[n_ /; n>8] := a[n] = ((9-n)*a[n-8] + 2*a[n-6] - 6*a[n-5] + (12-2*n)*a[n-4] + (4*n-6)*a[n-2] + 2*a[n-1])/(n+1); Table[a[n], {n, 2, 40}] (* Jean-François Alcover, Dec 02 2016, after Alois P. Heinz *)

Formula

G.f.: g(z)=(1+z)(z^4+2z^3+2z^2-1+Q)/(2z(1-z-z^2-z^3)), where Q = sqrt((1-z^2)(1-z-z^2-z^3)(1+z-z^2+z^3)).

Conjecture D-finite with recurrence (n+1)*a(n) +2*(-1)*a(n-1) +2*(-2*n+3)*a(n-2) +2*(n-6)*a(n-4) +6*(1)*a(n-5) -2*a(n-6) +(n-9)*a(n-8)=0. - R. J. Mathar, Jul 22 2022

a(n) ~ sqrt(2*r*(2-3*r)) * (25 + 18*r + 13*r^2) * (1 + r + r^2)^n / (22*sqrt(Pi*n)), where r = A192918. - Vaclav Kotesovec, Mar 08 2023