A273347 - OEIS

%I #25 Aug 19 2017 23:21:36

%S 1,4,16,60,218,778,2744,9600,33391,115638,399137,1374050,4720272,

%T 16187632,55434424,189607406,647872199,2211794804,7545239986,

%U 25722676402,87641357150,298456587038,1015914399813,3456670564220,11757143968393,39976448714086,135887220346719

%N The sum of the areas of the bargraphs of semiperimeter n (n>=2).

%D A. Blecher, C. Brennan, and A. Knopfmacher, Combinatorial parameters in bargraphs (preprint).

%H Alois P. Heinz, <a href="/A273347/b273347.txt">Table of n, a(n) for n = 2..1000</a>

%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 M. Bousquet-Mélou and R. Brak, <a href="https://hal.archives-ouvertes.fr/hal-00342024">Exactly solved models of polyominoes and polygons</a>, Chapter 3 of Polygons, Polyominoes and Polycubes, Lecture Notes in Physics, Vol. 775, 43-78, Springer, Berlin, Heidelberg 2009.

%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) = z^2 (2 - z - z^3 - z q )^2 /(1 - 4z + z^4 + q + z^2 q)^2, where q = sqrt(1 - 4z + 2z^2 + z^4) (see Section 4.3 of the Blecher et al. reference).

%F a(n) = Sum(k*A273346(n,k), k>=1).

%F a(n) = ((69-115*n+28*n^2)*a(n-1) -(264-265*n+52*n^2)*a(n-2) +(3*(29-29*n+4*n^2))*a(n-3) -(3*(10-21*n+4*n^2))*a(n-4) +(4*n-9) * (5*n-29)*a(n-5) +(4*n-13)*(n-6)*a(n-6) +(n-7)*(4*n-9)*a(n-7))/ (n*(4*n-13)) for n>=7. - _Alois P. Heinz_, Jun 04 2016

%e a(4) = 16  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 areas is 3 + 3 + 3 + 4 + 3 = 16.

%p Q := sqrt(1-4*z+2*z^2+z^4): g := z^2*(2-z-z^3-z*Q)^2/(1-4*z+z^4+Q+z^2*(2+Q))^2; gser:= series(g,z = 0,40): seq(coeff(gser, z, m), m = 2 .. 35);

%p # second Maple program:

%p a:= proc(n) option remember;

%p       `if`(n<7, [0$2, 1, 4, 16, 60, 218, 778][n+1],

%p       ((69-115*n+28*n^2)*a(n-1) -(264-265*n+52*n^2)*a(n-2)

%p        +(3*(29-29*n+4*n^2))*a(n-3) -(3*(10-21*n+4*n^2))*a(n-4)

%p        +(4*n-9)*(5*n-29)*a(n-5) +(4*n-13)*(n-6)*a(n-6)

%p        +(n-7)*(4*n-9)*a(n-7))/(n*(4*n-13)))

%p     end:

%p seq(a(n), n=2..30);  # _Alois P. Heinz_, Jun 04 2016

%t b[n_, y_, t_] := b[n, y, t] = Expand[If[n==0, 1-t, If[t<0, 0, b[n-1, y+1, 1]] + If[t>0 || y<2, 0, b[n, y-1, -1]] + If[y<1, 0, b[n-1, y, 0]*z^y]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 1, Exponent[p, z]}]][b[n, 0, 0]]; a[n_] := (row = T[n]; row.Range[Length[row]]); Table[a[n], {n, 2, 30}] (* _Jean-François Alcover_, Nov 29 2016 after _Alois P. Heinz_'s Maple code for A273346 *)

%Y Cf. A273346, A273348.

%K nonn

%O 2,2

%A _Emeric Deutsch_, Jun 03 2016