A276068 Sum of the lengths of the first descents in all bargraphs having semiperimeter n (n>=2). A descent is a maximal sequence of consecutive down steps.

1, 3, 9, 26, 74, 210, 598, 1715, 4963, 14504, 42808, 127553, 383451, 1162134, 3548060, 10904023, 33708595, 104756233, 327086895, 1025603074, 3228082910, 10195295005, 32300276271, 102622734570, 326893843104, 1043767139218, 3340051490096

Offset

2,2

Links

Table of n, a(n) for n=2..28.

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)(1-2z-z^2-Q)/(2z(1-2z)), where Q = sqrt((1-z)(1-3z-z^2-z^3)).

a(n) = Sum(k*A276067(n,k), k>=1).

Conjecture D-finite with recurrence (n+1)*a(n) +(-6*n+1)*a(n-1) +(10*n-13)*a(n-2) +(-4*n+13)*a(n-3) +(n-4)*a(n-4) +2*(-n+6)*a(n-5)=0. - R. J. Mathar, Jul 22 2022

Example

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 drawings show that the sum of the lengths of their first descents is 1+2+1+2+3.

Maple

g := (1/2)*(1-z)*(1-2*z-z^2-sqrt((1-z)*(1-3*z-z^2-z^3)))/(z*(1-2*z)): gser := series(g, z = 0, 38): seq(coeff(gser, z, n), n = 2 .. 35);

Mathematica

G = (1/2)(1-z)(1 - 2z - z^2 - Sqrt[(1-z)(1 - 3z - z^2 - z^3)])/(z(1-2z)) + O[z]^29;
Drop[CoefficientList[G, z], 2] // Flatten (* Jean-François Alcover, Aug 07 2018 *)

Crossrefs

Cf. A082582, A276067.

Sequence in context: A116423 A077845 A291000 * A171277 A289806 A303976

Adjacent sequences: A276065 A276066 A276067 * A276069 A276070 A276071

Keyword

nonn

Author

Emeric Deutsch, Aug 25 2016

Status

approved