

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



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
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OFFSET

2,2


LINKS

Table of n, a(n) for n=2..33.
M. BousquetMélou and A. Rechnitzer, The siteperimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86112.
Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088, 2016


FORMULA

G.f.: g(z) = (13z+3z^2  Q)/(2z(1z)), where Q = sqrt((13z+z^2)(13z+5z^24z^3)).


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 := ((13*z+3*z^2sqrt((13*z+z^2)*(13*z+5*z^24*z^3)))*(1/2))/(z*(1z)): 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(1z));
Drop[CoefficientList[g[z] + O[z]^(terms+2), z], 2] (* JeanFrançois Alcover, Aug 07 2018 *)


CROSSREFS

Cf. A082582, A023432.
Sequence in context: A078495 A161701 A038504 * A018405 A018419 A220826
Adjacent sequences: A275445 A275446 A275447 * A275449 A275450 A275451


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Sergi Elizalde, Aug 26 2016


STATUS

approved



