A273714 -id:A273714 - cp's OEIS frontend

A273713 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k doublerises (n>=2, k>=0). A doublerise in a bargraph is any pair of adjacent up steps.

1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 8, 13, 9, 4, 1, 17, 32, 28, 14, 5, 1, 37, 80, 81, 50, 20, 6, 1, 82, 201, 231, 165, 80, 27, 7, 1, 185, 505, 653, 526, 295, 119, 35, 8, 1, 423, 1273, 1824, 1644, 1036, 483, 168, 44, 9, 1, 978, 3217, 5058, 5034, 3535, 1848, 742, 228, 54, 10, 1

Offset: 2

Emeric Deutsch, May 28 2016

Number of entries in row n is n-1.
Sum of entries in row n = A082582(n).
T(n,0) = A004148(n-1) (the 2ndary structure numbers).
T(n,1) = A110320(n-2).
Sum(k*T(n,k), k>=0) = A273714(n).

			Row 4 is 2,2,1 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 0, 0, 1, 1, 2 doublerises.
Triangle starts
1;
1,1;
2,2,1;
4,5,3,1;
8,13,9,4,1

Alois P. Heinz, Rows n = 2..150, 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

Cf. A004148, A082582, A110320, A273714.

eq := z*G^2-(1-z-t*z-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 .. n-2) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y, t) option remember; expand(`if`(n=0, (1-t),
      `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`(y<1, 0, b(n-1, y, 0))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..n-2))(b(n, 0$2)):
seq(T(n), n=2..16);  # Alois P. Heinz, Jun 06 2016

b[n_, y_, t_] := b[n, y, t] = Expand[If[n == 0, 1 - t, 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[y < 1, 0, b[n - 1, y, 0]]]];
T[n_] := Function [p, Table[Coefficient[p, z, i], {i, 0, n - 2}]][b[n, 0, 0]];
Table[T[n], {n, 2, 16}] // Flatten (* Jean-François Alcover, Jul 29 2016, after Alois P. Heinz *)

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

A273896 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k UHU configurations, where U=(0,1), H(1,0); (n>=2, k>=0).

Original entry on oeis.org

1, 2, 4, 1, 9, 4, 22, 12, 1, 56, 35, 6, 146, 104, 24, 1, 388, 312, 86, 8, 1048, 938, 300, 40, 1, 2869, 2824, 1032, 170, 10, 7942, 8520, 3502, 680, 60, 1, 22192, 25763, 11748, 2632, 295, 12, 62510, 78064, 39072, 9926, 1330, 84, 1, 177308, 236976, 129100, 36640, 5712, 469, 14, 506008, 720574, 424344, 132960, 23660, 2352, 112, 1

Offset: 2

Emeric Deutsch, Jun 02 2016

Sum of entries in row n = A082582(n).
T(n,0) = A091561(n-1).
Sum(k*T(n,k), k>=0) = A273714(n-1). This implies that the number of UHUs in all bargraphs of semiperimeter n is equal to the number of doublerises in all bargraphs of semiperimeter n-1.

			Row 4 is [4,1] 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 0,1,0,0,0 UHU's.
Triangle starts
1;
2;
4,1;
9,4;
22,12,1;
56,35,6.

Alois P. Heinz, Rows n = 2..200, 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

Cf. A091561, A082582, A273714.

eq := z*G^2-(1-2*z-t*z^2)*G+z^2 = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 20)): for n from 2 to 18 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 2 to 18 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y, t, h) option remember; expand(
      `if`(n=0, (1-t), `if`(t<0, 0, b(n-1, y+1, 1, 0)*z^h)+
      `if`(t>0 or y<2, 0, b(n, y-1, -1, 0))+
      `if`(y<1, 0, b(n-1, y, 0, `if`(t>0, 1, 0)))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$3)):
seq(T(n), n=2..22); # Alois P. Heinz, Jun 06 2016

b[n_, y_, t_, h_] := b[n, y, t, h] = Expand[If[n==0, 1-t, If[t<0, 0, b[n-1, y+1, 1, 0]*z^h] + If[t>0 || y<2, 0, b[n, y-1, -1, 0]] + If[y<1, 0, b[n-1, y, 0, If[t>0, 1, 0]]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0, 0, 0]]; Table[T[n], {n, 2, 22}] // Flatten (* Jean-François Alcover, Dec 02 2016 after Alois P. Heinz *)

G.f.: G=G(t,z), where t marks number of UHU's and z marks semiperimeter, satisfies zG^2-(1-2z-tz^2)G+z^2 = 0.

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.

Original entry on oeis.org

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

Offset: 2

Emeric Deutsch and Sergi Elizalde, Aug 25 2016

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).

			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.

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

Cf. A023431, A025246, A082582, A273714.

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

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 *)

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.

cp's OEIS Frontend

A273713 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k doublerises (n>=2, k>=0). A doublerise in a bargraph is any pair of adjacent up steps.

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A273896 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k UHU configurations, where U=(0,1), H(1,0); (n>=2, k>=0).

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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.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula