A275448 The number of weakly alternating bargraphs of semiperimeter n. A bargraph is said to be weakly alternating if its ascents and descents alternate. An ascent (descent) is a maximal sequence of consecutive U (D) steps.

1, 2, 3, 4, 6, 12, 28, 65, 146, 327, 749, 1756, 4165, 9913, 23652, 56687, 136627, 330969, 804915, 1963830, 4805523, 11793046, 29019930, 71589861, 177006752, 438561959, 1088714711, 2707615555, 6745272783, 16830750107, 42058592797, 105248042792

Offset

2,2

Links

Table of n, a(n) for n=2..33.

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) = (1-3z+3z^2 - Q)/(2z(1-z)), where Q = sqrt((1-3z+z^2)(1-3z+5z^2-4z^3)).

D-finite with recurrence (n+1)*a(n) +(-7*n+2)*a(n-1) +3*(7*n-11)*a(n-2) +(-37*n+107)*a(n-3) +3*(13*n-54)*a(n-4) +3*(-7*n+37)*a(n-5) +2*(2*n-13)*a(n-6)=0. - R. J. Mathar, Jul 22 2022

Example

a(4)=3 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 only [1,1,1],[2,2], and [3] lead to weakly alternating bargraphs.

Maple

g := ((1-3*z+3*z^2-sqrt((1-3*z+z^2)*(1-3*z+5*z^2-4*z^3)))*(1/2))/(z*(1-z)): gser:= series(g, z=0, 43): seq(coeff(gser, z, n), n=2..40);

Mathematica

terms = 32;
g[z_] = ((1 - 3z + 3z^2 - Sqrt[(1 - 3z + z^2)(1 - 3z + 5z^2 - 4z^3)])*(1/2) )/(z(1-z));
Drop[CoefficientList[g[z] + O[z]^(terms+2), z], 2] (* Jean-François Alcover, Aug 07 2018 *)

Crossrefs

Cf. A082582, A023432.

Sequence in context: A161701 A038504 A352904 * A018405 A018419 A220826

Adjacent sequences: A275445 A275446 A275447 * A275449 A275450 A275451

Keyword

nonn

Author

Emeric Deutsch, Sergi Elizalde, Aug 26 2016

Status

approved