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%I #38 Jul 26 2022 13:04:42
%S 0,1,4,14,47,155,508,1662,5438,17809,58395,191732,630373,2075221,
%T 6840140,22571800,74564874,246568051,816099650,2703492238,8963064935,
%U 29738123605,98735734915,328034119098,1090509180192,3627343273885,12072071392105,40197107361740,133910579452363
%N Number of doublerises in all bargraphs having semiperimeter n (n>=2). A doublerise in a bargraph is any pair of adjacent up steps.
%C 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
%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 and S. Elizalde, <a href="http://arxiv.org/abs/1609.00088">Statistics on bargraphs viewed as cornerless Motzkin paths</a>, arXiv preprint arXiv:1609.00088 [math.CO], 2016.
%F G.f.: g = (1 - 2z - z^2 - Q)/(2Q), where Q = sqrt(1 - 4z + 2z^2 + z^4).
%F a(n) = Sum_{k>0} k*A273713(n,k).
%F From _Benedict W. J. Irwin_, May 29 2016: (Start)
%F Let y(0)=1, y(1)=2, y(2)=5, y(3)=14,
%F Let (n+2)*y(n) + (2*n+6)*y(n+2) - (4*n+14)*y(n+3) + (n+4)*y(n+4)=0,
%F a(n) = (y(n+2)-2*y(n+1)-y(n))/2.
%F (End)
%F 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
%e 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.
%p 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);
%t 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 *)
%Y Cf. A082582, A273713, A271943.
%K nonn
%O 2,3
%A _Emeric Deutsch_, May 28 2016
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