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 A274494 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k as the length of the longest initial sequence of the form UHUH...  (n>=2, 1<=k<=2*floor(n/2). 1
 0, 1, 1, 1, 3, 1, 0, 1, 8, 2, 1, 2, 22, 5, 4, 3, 0, 1, 62, 13, 12, 6, 1, 3, 178, 35, 35, 15, 5, 6, 0, 1, 519, 97, 103, 40, 17, 13, 1, 4, 1533, 275, 306, 110, 53, 33, 6, 10, 0, 1, 4578, 794, 917, 310, 163, 90, 23, 24, 1, 5, 13800, 2327, 2770, 891, 501, 253, 77, 63, 7, 15, 0, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,5 COMMENTS Number of entries in row n is 2*floor(n/2). Sum of entries in row n = A082582(n). Sum(k*T(n,k),k>=0) = A274495(n). LINKS 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 [math.CO], 2016. FORMULA G.f.: G = G(t,z) satisfies aG^2 + bG + c = 0, where a = z(1-t^2*z-t^2*z^3+t^4*z^3), b = -t(1-3z+z^2+tz^2-t^2*z^2-z^3+2t^2*z^3+tz^4-2t^3*z^4+t^2*z^4), c = t^2*z^2*(t+z-2tz-tz^2+t^2*z^2). EXAMPLE Row 4 is 3,1,0,1 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 the lengths of the longest initial sequence of the form UHUH... are 2,4,1,1,1, respectively. Triangle starts 0,1; 1,1; 3,1,0,1; 8,2,1,2; 22,5,4,3,0,1; MAPLE a := z*(1-t^2*z-t^2*z^3+t^4*z^3): b := -t*(1-3*z+z^2+t*z^2-t^2*z^2-z^3+2*t^2*z^3+t*z^4-2*t^3*z^4+t^2*z^4): c := t^2*z^2*(t+z-2*t*z-t*z^2+t^2*z^2): eq := a*G^2+b*G+c = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 21)): for n from 2 to 18 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 2 to 18 do seq(coeff(P[n], t, j), j = 1 .. 2*floor((1/2)*n)) end do; # yields sequence in triangular form MATHEMATICA nmax = 12; a = z (1 - t^2 z - t^2 z^3 + t^4 z^3); b = -t (1 - 3z + z^2 + t z^2 - t^2 z^2 - z^3 + 2t^2 z^3 + t z^4 - 2t^3 z^4 + t^2 z^4); c = t^2 z^2 (t + z - 2t z - t z^2 + t^2 z^2); G = 0; Do[G = Series[(-c - a G^2)/b, {z, 0, nmax}, {t, 0, nmax}] // Normal, {nmax}]; cc = CoefficientList[G, z]; row[n_] := CoefficientList[cc[[n+1]], t] // Rest; Table[row[n], {n, 2, nmax}] // Flatten (* Jean-François Alcover, Jul 25 2018 *) CROSSREFS Cf. A082582, A274495. Sequence in context: A048993 A264431 A257050 * A274490 A193357 A112413 Adjacent sequences:  A274491 A274492 A274493 * A274495 A274496 A274497 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Sergi Elizalde, Aug 26 2016 STATUS approved

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Last modified June 7 01:26 EDT 2020. Contains 334836 sequences. (Running on oeis4.)