A273898 Sum of the abscissae of the first descents of all bargraphs of semiperimeter n (n>=2).

1, 3, 9, 27, 81, 244, 739, 2251, 6895, 21232, 65703, 204245, 637573, 1997892, 6282635, 19820580, 62716923, 198997349, 633015543, 2018391204, 6449819095, 20652628601, 66256638509, 212939343591, 685497649231, 2210217592624, 7136781993563, 23076554161563

Offset: 2

Author

Emeric Deutsch, Jun 06 2016

Keywords

• nonn

Comments

A descent in a bargraph is a maximal sequence of adjacent down steps.

Examples

			a(4)=9 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1],[1,2],[2,1],[2,2],[3] and the corresponding pictures give the values 3,2,1,2,1 for the abscissae of their first descents.
		

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

Crossrefs

Cf. A082582, A273897.

Programs

Maple

g := ((1-4*z+3*z^2-(1-2*z)*Q)*(1/2))/z^3: Q := sqrt(1-4*z+2*z^2+z^4): gser := series(g,z = 0,40): seq(coeff(gser, z, n), n = 2 .. 35);
# second Maple program:
a:= proc(n) option remember; `if`(n<4, [0$2, 1, 3][n+1],
     ((2*(14*n^2+6+13*n))*a(n-1)-(2*(7*n^2-6-4*n))*a(n-2)
     +12*a(n-3) -(n-4)*(3+7*n)*a(n-4))/((n+3)*(7*n-4)))
    end:
seq(a(n), n=2..40);  # Alois P. Heinz, Jun 07 2016

Mathematica

a[n_] := a[n] = If[n<4, {0, 0, 1, 3}[[n+1]], ((2*(14*n^2+6+13*n))*a[n-1] - (2*(7*n^2-6-4*n))*a[n-2] + 12*a[n-3] - (n-4)*(3+7*n)*a[n-4])/((n+3)*(7*n - 4))]; Table[a[n], {n, 2, 40}] (* Jean-François Alcover, Dec 02 2016 after Alois P. Heinz *)

Formula

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

a(n) = Sum(k*A273897(n,k), k>=1).

a(n) = A082582(n+2)-2*A082582(n+1).

D-finite with recurrence (n+3)*a(n) +2*(-3*n-4)*a(n-1) +2*(5*n-2)*a(n-2) +4*(-n+2)*a(n-3) +(n-3)*a(n-4) +2*(-n+5)*a(n-5)=0. - R. J. Mathar, Jul 24 2022