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A273347 The sum of the areas of the bargraphs of semiperimeter n (n>=2). 4
1, 4, 16, 60, 218, 778, 2744, 9600, 33391, 115638, 399137, 1374050, 4720272, 16187632, 55434424, 189607406, 647872199, 2211794804, 7545239986, 25722676402, 87641357150, 298456587038, 1015914399813, 3456670564220, 11757143968393, 39976448714086, 135887220346719 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,2
REFERENCES
A. Blecher, C. Brennan, and A. Knopfmacher, Combinatorial parameters in bargraphs (preprint).
LINKS
M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.
M. Bousquet-Mélou and R. Brak, Exactly solved models of polyominoes and polygons, Chapter 3 of Polygons, Polyominoes and Polycubes, Lecture Notes in Physics, Vol. 775, 43-78, Springer, Berlin, Heidelberg 2009.
Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088, 2016
FORMULA
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).
a(n) = Sum(k*A273346(n,k), k>=1).
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
EXAMPLE
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.
MAPLE
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);
# second Maple program:
a:= proc(n) option remember;
`if`(n<7, [0$2, 1, 4, 16, 60, 218, 778][n+1],
((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)))
end:
seq(a(n), n=2..30); # Alois P. Heinz, Jun 04 2016
MATHEMATICA
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 *)
CROSSREFS
Sequence in context: A089932 A120926 A255303 * A268939 A269635 A267928
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jun 03 2016
STATUS
approved

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Last modified June 15 18:41 EDT 2024. Contains 373410 sequences. (Running on oeis4.)