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A273904 Number of even-length columns in all bargraphs having semiperimeter n (n>=2).

%I A273904 #15 Aug 19 2017 23:20:42
%S A273904 0,1,4,13,44,149,498,1656,5498,18236,60456,200409,664464,2203755,
%T A273904 7311894,24271290,80605250,267821525,890305418,2961015981,9852481830,
%U A273904 32798011430,109229396466,363927233758,1213012655490,4044684629394,13491663770344
%N A273904 Number of even-length columns in all bargraphs having semiperimeter n (n>=2).
%H A273904 Alois P. Heinz, <a href="/A273904/b273904.txt">Table of n, a(n) for n = 2..1000</a>
%H A273904 M. Bousquet-Mélou and A. Rechnitzer, <a href="http://dx.doi.org/10.1016/S0196-8858(02)00553-5">The site-perimeter of bargraphs</a>, Adv. in Appl. Math. 31 (2003), 86-112.
%H A273904 Emeric Deutsch, S Elizalde, <a href="http://arxiv.org/abs/1609.00088">Statistics on bargraphs viewed as cornerless Motzkin paths</a>, arXiv preprint arXiv:1609.00088, 2016
%F A273904 G.f.:  g(z)=((1-z)(1-3z+z^2-z^3)-(1-z)^2*Q)/(2z(1+z^2)*Q), where Q = sqrt((1-z)(1-3z-z^2-z^3)).
%F A273904 a(n) = Sum(k*A273903(n,k), k>=0).
%e A273904 a(4) = 4 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1],[1,2],[2,1],[2,2],[3] and, clearly, they have 0,1,1,2,0 columns of even length.
%p A273904 Q := sqrt((1-z)*(1-3*z-z^2-z^3)): g := (((1-z)*(1-3*z+z^2-z^3)-(1-z)^2*Q)*(1/2))/(z*(1+z^2)*Q): gser := series(g, z = 0, 40): seq(coeff(gser, z, m), m = 2 .. 35);
%p A273904 # second Maple program:
%p A273904 a:= proc(n) option remember; `if`(n<7, [0$3, 1, 4, 13, 44]
%p A273904       [n+1], ((7*n-22)*(n-6)*a(n-7) -(5*n^2-21*n+6)*a(n-6)+
%p A273904       (21*n^2-180*n+404)*a(n-5) -(43*n^2-265*n+332)*a(n-4)
%p A273904       +(41*n^2-226*n+308)*a(n-3) -(43*n^2-257*n+308)*a(n-2)
%p A273904       +(27*n^2-110*n+36)*a(n-1))/ ((n+1)*(5*n-18)))
%p A273904     end:
%p A273904 seq(a(n), n=2..40);  # _Alois P. Heinz_, Jun 24 2016
%t A273904 Q = Sqrt[(1-z)*(1-3*z-z^2-z^3)]; g = (((1-z)*(1-3*z+z^2-z^3) - (1-z)^2 * Q)*(1/2))/(z*(1+z^2)*Q); gser = g + O[z]^40; CoefficientList[gser, z][[3 ;; -1]] (* _Jean-François Alcover_, Oct 04 2016, adapted from Maple *)
%Y A273904 Cf. A082582, A273901, A273902, A273903.
%K A273904 nonn
%O A273904 2,3
%A A273904 _Emeric Deutsch_, Jun 23 2016