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A273714 Number of doublerises in all bargraphs having semiperimeter n (n>=2). A doublerise in a bargraph is any pair of adjacent up steps. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,3

LINKS

Table of n, a(n) for n=2..30.

M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.

Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088, 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)

Conjecture: n*(2*n-5)*(2*n-7)*a(n) -8*(n-1)*(n-2)*(2*n-7)*a(n-1) +2*(n-2)*(4*n^2-20*n+17)*a(n-2) +2*(2*n-3)*a(n-3) +(n-4)*(2*n-3)*(2*n-5)*a(n-4)=0. - R. J. Mathar, Jun 06 2016

Conjecture: n*(2*n-1)*a(n) +(n-1)*(2*n-49)*a(n-1) -2*(n-2)*(18*n-89)*a(n-2) +2*(10*n^2-69*n+102)*a(n-3) +(2*n^2-5*n+10)*a(n-4) +(10*n-29)*(n-5)*a(n-5)=0. - R. J. Mathar, Jun 06 2016

Conjecture: 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

EXAMPLE

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.

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 *)

CROSSREFS

Cf. A082582, A273713.

Sequence in context: A104487 A247210 A094789 * A082574 A289780 A320404

Adjacent sequences:  A273711 A273712 A273713 * A273715 A273716 A273717

KEYWORD

nonn

AUTHOR

Emeric Deutsch, May 28 2016

STATUS

approved

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Last modified April 20 16:23 EDT 2021. Contains 343135 sequences. (Running on oeis4.)