A273902 Number of odd-length columns in all bargraphs having semiperimeter n (n>=2).

1, 2, 6, 20, 64, 204, 656, 2120, 6873, 22350, 72881, 238232, 780384, 2561164, 8419766, 27721784, 91397927, 301710074, 997087170, 3298556716, 10922576840, 36199599880, 120068987717, 398547827336, 1323821438203, 4400043488826, 14633372199291

Offset

2,2

Links

Alois P. Heinz, Table of n, a(n) for n = 2..1000

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

Formula

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

a(n) = Sum(k*A273901(n,k), k>=0).

D-finite with recurrence n*(12*n-35)*a(n) -(n-1)*(55*n-178)*a(n-1) +2*(n-2)*(32*n-143)*a(n-2) +(-69*n^2+370*n-501)*a(n-3) +(64*n^2-443*n+524)*a(n-4) +(-21*n^2+215*n-526)*a(n-5) +4*(3*n+2)*(n-6)*a(n-6) -(7*n-29)*(n-7)*a(n-7)=0. - R. J. Mathar, Jul 26 2022

Example

a(4) = 6 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1],[1,2],[2,1],[2,2],[3] and, clearly, they have 3,1,1,0,1 columns of odd length.

Maple

Q := sqrt((1-z)*(1-3*z-z^2-z^3)): g := (((1-z)*(1-z-z^2-z^3)-(1-z^2)*Q)*(1/2))/((1+z^2)*Q): gser := series(g, z = 0, 40): seq(coeff(gser, z, m), m = 2 .. 35);
# second Maple program:
a:= proc(n) option remember; `if`(n<7, [0$2, 1, 2, 6, 20, 64]
       [n+1], ((n-1)*(55*n-178)*a(n-1)-(2*(n-2))*(32*n-143)*
       a(n-2)+(501-370*n+69*n^2)*a(n-3)-(524-443*n+64*n^2)*
       a(n-4)+(526-215*n+21*n^2)*a(n-5)-(4*(3*n+2))*(n-6)*
       a(n-6)+(n-7)*(-29+7*n)*a(n-7))/ (n*(12*n-35)))
    end:
seq(a(n), n=2..35);  # Alois P. Heinz, Jun 23 2016

Mathematica

Q = Sqrt[(1-z)*(1-3*z-z^2-z^3)]; g = (((1-z)*(1-z-z^2-z^3) - (1-z^2)*Q)*(1/2))/((1+z^2)*Q); gser = g + O[z]^40; CoefficientList[gser, z][[3 ;; -1]] (* Jean-François Alcover, Oct 04 2016, adapted from Maple *)

Crossrefs

Cf. A082582, A273901, A273903, A273904.

Sequence in context: A199102 A053730 A220874 * A181301 A302612 A005726

Adjacent sequences: A273899 A273900 A273901 * A273903 A273904 A273905

Keyword

nonn

Author

Emeric Deutsch, Jun 22 2016

Status

approved