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
Keywords
Examples
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.
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
Programs
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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 *)
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