A273351 Number of up steps in all bargraphs of semiperimeter n (n>=2).
1, 3, 10, 32, 102, 326, 1046, 3370, 10899, 35369, 115123, 375705, 1228970, 4028366, 13228516, 43511464, 143329157, 472761015, 1561246112, 5161512902, 17081176912, 56579333508, 187570898065, 622318325281, 2066208751201, 6864800067363, 22821993704857, 75915970992635, 252667993114760
Offset: 2
Keywords
Examples
a(4) = 10 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] which, clearly, have 1,2,2,2,3 up steps.
Links
- A. Blecher, C. Brennan and A. Knopfmacher, Combinatorial parameters in bargraphs, Quaestiones Mathematicae, 39 (2016), 619-635.
- M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.
Programs
-
Maple
g := ((1-z-z^2-z^3-(1+z)*sqrt(1-4*z+2*z^2+z^4))*(1/2))/sqrt(1-4*z+2*z^2+z^4): gser := series(g, z = 0, 33): seq(coeff(gser, z, n), n = 2 .. 30);
Formula
G.f.: g(z) = (1-z-z^2-z^3-(1+z)*h)/(2*h), where h = sqrt(1-4*z+2*z^2+z^4).
a(n) = Sum_{k>=1} k*A273350(n,k).
Conjecture: n*(13*n-40)*a(n) +(-55*n^2+211*n-129)*a(n-1) +(38*n^2-212*n+255)*a(n-2) +(-6*n^2+56*n-75)*a(n-3) +(13*n^2-66*n+3)*a(n-4) -(3*n-13)*(n-6)*a(n-5)=0. - R. J. Mathar, Jun 06 2016
Conjecture: n*(n-3)*(n-2)^2*a(n) -(n-3)*(2*n-3)*(2*n^2-6*n+3) *a(n-1) +(2*n^4-16*n^3+41*n^2-36*n+5) *a(n-2) +2*(n-1)*(2*n-5) *a(n-3) +(n-2)*(n-5)*(n-1)^2 *a(n-4)=0. - R. J. Mathar, Jun 06 2016