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A276066 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having a total of k double rises and double falls (n>=2,k>=0). A double rise (fall) in a bargraph is any pair of adjacent up (down) steps. 1
1, 1, 0, 1, 1, 2, 1, 0, 1, 2, 4, 1, 4, 1, 0, 1, 4, 6, 8, 8, 1, 6, 1, 0, 1, 7, 14, 22, 12, 19, 12, 1, 8, 1, 0, 1, 13, 34, 43, 48, 55, 18, 35, 16, 1, 10, 1, 0, 1, 26, 72, 105, 148, 109, 116, 103, 24, 56, 20, 1, 12, 1, 0, 1, 52, 154, 276, 344, 347, 398, 205, 232, 166, 30, 82, 24, 1, 14, 1, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,6

COMMENTS

Number of entries in row n is 2n-3.

Sum of entries in row n = A082582(n).

T(n,0) = A023431(n-2) = A025246(n+1).

Sum(k*T(n,k),k>=0) = 2*A273714(n).

LINKS

Alois P. Heinz, Rows n = 2..140, flattened

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 = G(t,z) satisfies zG^2 - (1-z - t^2*z - 2tz^2+t^2*z^2)G + z^2 = 0.

The g.f. B(t,s,z) of bargraphs, where t(s) marks double rises (falls) and z marks semiperimeter, satisfies zB^2 - (1-(1+ts)z +(ts- t-s)z^2)B + z^2 = 0.

EXAMPLE

Row 4 is 1,2,1,0,1 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 they have a total of  0, 1, 1, 2, 4 double rises and double falls, respectively.

Triangle starts:

1;

1,0,1;

1,2,1,0,1;

2,4,1,4,1,0,1;

4,6,8,8,1,6,1,0,1.

MAPLE

eq := z*G^2-(1-z-t^2*z-2*t*z^2+t^2*z^2)*G+z^2 = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 22)): for n from 2 to 20 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 2 to 20 do seq(coeff(P[n], t, j), j = 0 .. 2*n-4) end do; # yields sequence in triangular form.

# second Maple program:

b:= proc(n, y, t) option remember; expand(`if`(n=0, (1-t)*

      z^(y-1), `if`(t<0, 0, b(n-1, y+1, 1)*`if`(t=1, z, 1))+

     `if`(t>0 or y<2, 0, b(n, y-1, -1)*`if`(t=-1, z, 1))+

     `if`(y<1, 0, b(n-1, y, 0))))

    end:

T:= n->(p->seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$2)):

seq(T(n), n=2..12);  # Alois P. Heinz, Aug 25 2016

MATHEMATICA

b[n_, y_, t_] := b[n, y, t] = Expand[If[n == 0, (1 - t)*z^(y - 1), If[t < 0, 0, b[n - 1, y + 1, 1]*If[t == 1, z, 1]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1]*If[t == -1, z, 1]] + If[y < 1, 0, b[n - 1, y, 0]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0, 0]]; Table[T[n], {n, 2, 12}] // Flatten (* Jean-François Alcover, Dec 02 2016 after Alois P. Heinz *)

CROSSREFS

Cf. A023431, A025246, A082582, A273714.

Sequence in context: A248157 A290428 A136868 * A145895 A114503 A103528

Adjacent sequences:  A276063 A276064 A276065 * A276067 A276068 A276069

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch and Sergi Elizalde, Aug 25 2016

STATUS

approved

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Last modified March 26 12:43 EDT 2019. Contains 321497 sequences. (Running on oeis4.)