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%I #18 Nov 08 2016 19:58:29
%S 1,3,10,32,102,326,1046,3370,10899,35369,115123,375705,1228970,
%T 4028366,13228516,43511464,143329157,472761015,1561246112,5161512902,
%U 17081176912,56579333508,187570898065,622318325281,2066208751201,6864800067363,22821993704857,75915970992635,252667993114760
%N Number of up steps in all bargraphs of semiperimeter n (n>=2).
%H A. Blecher, C. Brennan and A. Knopfmacher, <a href="http://www.tandfonline.com/doi/abs/10.2989/16073606.2015.1121932">Combinatorial parameters in bargraphs</a>, Quaestiones Mathematicae, 39 (2016), 619-635.
%H 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.
%F G.f.: g(z)=(1-z-z^2-z^3-(1+z)h)/2h), where h = sqrt(1-4z+2z^2+z^4).
%F a(n) = Sum(k*A273350(n,k), k>=1).
%F Conjecture: n*(13*n-40)*a(n) +(-55*n^2+211*n-129)*a(n-1) +(38*n^2-212*n+255)*a(n-2) +(-6*n^2+56*n-75)*a(n-3) +(13*n^2-66*n+3)*a(n-4) -(3*n-13)*(n-6)*a(n-5)=0. - _R. J. Mathar_, Jun 06 2016
%F Conjecture: n*(n-3)*(n-2)^2*a(n) -(n-3)*(2*n-3)*(2*n^2-6*n+3) *a(n-1) +(2*n^4-16*n^3+41*n^2-36*n+5) *a(n-2) +2*(n-1)*(2*n-5) *a(n-3) +(n-2)*(n-5)*(n-1)^2 *a(n-4)=0. - _R. J. Mathar_, Jun 06 2016
%e a(4) = 10 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] which, clearly, have 1,2,2,2,3 up steps.
%p g := ((1-z-z^2-z^3-(1+z)*sqrt(1-4*z+2*z^2+z^4))*(1/2))/sqrt(1-4*z+2*z^2+z^4): gser := series(g, z = 0, 33): seq(coeff(gser, z, n), n = 2 .. 30);
%Y Cf. A082582, A273350.
%K nonn
%O 2,2
%A _Emeric Deutsch_, Jun 02 2016
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