A273714 Number of doublerises in all bargraphs having semiperimeter n (n>=2). A doublerise in a bargraph is any pair of adjacent up steps.

0, 1, 4, 14, 47, 155, 508, 1662, 5438, 17809, 58395, 191732, 630373, 2075221, 6840140, 22571800, 74564874, 246568051, 816099650, 2703492238, 8963064935, 29738123605, 98735734915, 328034119098, 1090509180192, 3627343273885, 12072071392105, 40197107361740, 133910579452363

Offset: 2

Author

Emeric Deutsch, May 28 2016

Keywords

• nonn

Comments

a(n) appears to be the number of 021-avoiding ascent sequences (A022493) with exactly one repeated nonzero entry, where repeated means two consecutive equal entries. For example, a(4) = 4 counts 0011, 0110, 0112, 0122, and a(5) = 14 counts 00011, 00110, 00112, 00122, 01011, 01022, 01100, 0110 1, 01102, 01120, 01123, 0122 0, 01223, 01233. - David Callan, Nov 21 2021

Examples

			a(4) = 4 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 drawings show that they have 0, 0, 1, 1, 2 doublerises.
		

Links

• M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.
• Emeric Deutsch and S. Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088 [math.CO], 2016.

Crossrefs

Cf. A082582, A273713, A271943.

Programs

Maple

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

Mathematica

F[k_] := DifferenceRoot[Function[{y, n}, {(2 + n) y[n] + (6 + 2 n) y[2 + n] + (-14 - 4 n) y[3 + n] + (4 + n) y[4 + n] == 0, y[0] == 1, y[1] == 2, y[2] == 5, y[3] == 14}]][k]; Table[1/2 (-F[n] - 2 F[n + 1] + F[n + 2]), {n, 0, 20}] (* Benedict W. J. Irwin, May 29 2016 *)

Formula

G.f.: g = (1 - 2z - z^2 - Q)/(2Q), where Q = sqrt(1 - 4z + 2z^2 + z^4).

a(n) = Sum_{k>0} k*A273713(n,k).

From Benedict W. J. Irwin, May 29 2016: (Start)

Let y(0)=1, y(1)=2, y(2)=5, y(3)=14,

Let (n+2)*y(n) + (2*n+6)*y(n+2) - (4*n+14)*y(n+3) + (n+4)*y(n+4)=0,

a(n) = (y(n+2)-2*y(n+1)-y(n))/2.

(End)

D-finite with recurrence n*a(n) +6*(-n+1)*a(n-1) +9*(n-2)*a(n-2) -6*a(n-3) +(-n+8) * a(n-4) +2*(-n+4)*a(n-5) +(-n+6)*a(n-6)=0. - R. J. Mathar, Jun 06 2016