%I #21 Aug 19 2017 23:07:07
%S 1,3,9,26,75,218,640,1898,5682,17155,52187,159827,492417,1525222,
%T 4746906,14837444,46558573,146614539,463186317,1467631144,4662899110,
%U 14851847390,47414162252,151692982789,486280700344,1561757802585,5024492606869,16191028967145,52253656263073,168880350860512
%N The sum of the first-column lengths of all bargraphs of semiperimeter n (n>=2).
%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.
%H 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 G.f.: G = (z^6 + z^5 + z^4 - 2z^3 - 5z^2 +5z - 1 - (z^3 + 2z^2 + 2z - 1)Z)/(2 Z z^2), where Z = (1 - z)sqrt((1 - z)(1 - 3z - z^2 - z^3)).
%F a(n) = Sum(k*A273342(n,k), k>=1).
%F Conjecture: (n+2)*a(n) +(-5*n-3)*a(n-1) +(5*n-2)*a(n-2) +(2*n-9)*a(n-3) +(-n+3)*a(n-4) +(-n+4)*a(n-5) +(-n+7)*a(n-6)=0. - _R. J. Mathar_, Jun 02 2016
%e 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, clearly, the sum of their first-columns lengths is 1+1+2+2+3=9.
%p Z := (1-z)*sqrt((1-z)*(1-3*z-z^2-z^3)): g := ((z^6+z^5+z^4-2*z^3-5*z^2+5*z-1-(z^3+2*z^2+2*z-1)*Z)*(1/2))/(z^2*Z): gser := series(g,z=0,44): seq(coeff(gser,z,n),n=2..40);
%t Drop[CoefficientList[Series[(x^6 + x^5 + x^4 - 2 x^3 - 5 x^2 + 5 x - 1 - (x^3 + 2 x^2 + 2 x - 1) #)/(2 # x^2) &[(1 - x) Sqrt[(1 - x) (1 - 3 x - x^2 - x^3)]], {x, 0, 31}], x], 2] (* _Michael De Vlieger_, May 21 2016 *)
%Y Cf. A082582, A273342.
%K nonn
%O 2,2
%A _Emeric Deutsch_, May 21 2016