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A273897 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having abscissa of first descent k (n>=2, 1<=k<=n-1).

%I A273897 #21 Jul 24 2018 09:40:51
%S A273897 1,1,1,2,2,1,4,5,3,1,9,12,9,4,1,22,30,25,14,5,1,57,78,69,44,20,6,1,
%T A273897 154,210,192,133,70,27,7,1,429,582,542,396,230,104,35,8,1,1223,1651,
%U A273897 1554,1176,731,369,147,44,9,1,3550,4772,4521,3504,2285,1248,560,200,54,10,1
%N A273897 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having abscissa of first descent k (n>=2, 1<=k<=n-1).
%C A273897 Number of entries in row n is n-1.
%C A273897 Sum of entries in row n = A082582(n).
%C A273897 T(n,1) = A105633(n-3) (n>=3).
%C A273897 Sum(k*T(n,k), k>=1) = A273898(n).
%H A273897 Alois P. Heinz, <a href="/A273897/b273897.txt">Rows n = 2..150, flattened</a>
%H A273897 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 A273897 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 A273897 G.f.: G(t,z)=(1/2)tz(1-2tz-z^2-sqrt(1-4z+2z^2+z^4))/(1-t-z+t^2z+tz^4), where z marks semiperimeter and t marks the abscissa of the first descent.
%e A273897 Row 4 is 2,2,1 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 pictures give the values 3,2,1,2,1 for the abscissae of the first descents.
%e A273897 Triangle starts
%e A273897 1;
%e A273897 1,1;
%e A273897 2,2,1;
%e A273897 4,5,3,1;
%e A273897 9,12,9,4,1;
%e A273897 22,30,25,14,5,1.
%p A273897 G := (1/2)*t*z*(1-2*t*z-z^2-sqrt(1-4*z+2*z^2+z^4))/(1-t-z+t^2*z+t*z^2): Gser := simplify(series(G, z = 0, 20)): for n from 2 to 18 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 2 to 18 do seq(coeff(P[n], t, j), j = 1 .. n-1) end do; # yields sequence in triangular form
%t A273897 nmax = 13; G = (1/2) t z (1 - 2t z - z^2 - Sqrt[1 - 4z + 2z^2 + z^4])/(1 - t - z + t^2 z + t z^2); Gser = G + O[z]^nmax;
%t A273897 Do[P[n] = Expand[Coefficient[Gser, z, n]], {n, 2, nmax}];
%t A273897 Table[CoefficientList[P[n]/t, t], {n, 2, nmax}] // Flatten (* _Jean-François Alcover_, Jul 24 2018, from Maple *)
%Y A273897 Cf. A082582, A105633, A273898.
%K A273897 nonn,tabl
%O A273897 2,4
%A A273897 _Emeric Deutsch_, Jun 06 2016