A274490 - OEIS

%I #19 Jul 25 2018 08:06:29

%S 0,1,1,0,1,3,1,0,1,8,3,1,0,1,22,8,3,1,0,1,62,22,8,3,1,0,1,178,62,22,8,

%T 3,1,0,1,519,178,62,22,8,3,1,0,1,1533,519,178,62,22,8,3,1,0,1,4578,

%U 1533,519,178,62,22,8,3,1,0,1,13800,4578,1533,519,178,62,22,8,3,1,0,1

%N Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n starting with k columns of length 1 (n>=2, k>=0).

%C Number of entries in row n is n.

%C Sum of entries in row n = A082582(n).

%C T(n,0) = A188464(n-3) (n>=3).

%C Sum_{k>=0} k*T(n,k) = A105633(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.: (1-3*z+z^2+2*t*z^3-z^3-(1-z)*sqrt((1-z)*(1-3*z-z^2-z^3)))/(2*z*(1-t*z)).

%e Row 4 is 3,1,0,1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] and, clearly, they start with 3, 1, 0, 0, 0 columns of length 1.

%e Triangle starts

%e 0,1;

%e 1,0,1;

%e 3,1,0,1;

%e 8,3,1,0,1;

%e 22,8,3,1,0,1

%p G := (1-3*z+z^2+2*t*z^3-z^3-(1-z)*sqrt((1-z)*(1-3*z-z^2-z^3)))/(2*z*(1-t*z)): Gser := simplify(series(G, z = 0, 22)): for n from 2 to 18 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 2 to 18 do seq(coeff(P[n], t, k), k = 0 .. n-1) end do; # yields sequence in triangular form

%t nmax = 12;

%t g = (1 - 3z + z^2 + 2t z^3 - z^3 - (1-z) Sqrt[(1-z)(1 - 3z - z^2 - z^3)])/ (2z (1 - t z));

%t cc = CoefficientList[g + O[z]^(nmax+1), z];

%t T[n_, k_] := Coefficient[cc[[n+1]], t, k];

%t Table[T[n, k], {n, 2, nmax}, {k, 0, n-1}] // Flatten (* _Jean-François Alcover_, Jul 25 2018 *)

%Y Cf. A082582, A105633, A188464.

%K nonn,tabf

%O 2,6

%A _Emeric Deutsch_, Jun 25 2016