A273720 -id:A273720 - cp's OEIS frontend

A273719 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k horizontal steps in the peaks (n>=2, k>=1).

1, 1, 1, 3, 1, 1, 8, 3, 1, 1, 21, 9, 3, 1, 1, 55, 27, 10, 3, 1, 1, 144, 82, 33, 11, 3, 1, 1, 377, 251, 110, 39, 12, 3, 1, 1, 987, 770, 368, 139, 45, 13, 3, 1, 1, 2584, 2358, 1229, 495, 169, 51, 14, 3, 1, 1, 6765, 7191, 4085, 1755, 632, 200, 57, 15, 3, 1, 1

Offset: 2

Emeric Deutsch, Jun 01 2016

Number of entries in row n is n-1.
Sum of entries in row n = A082582(n).
T(n,1) = A088305(n-2) = F(2n-4) where F(n) are the Fibonacci numbers  A000045.
Sum(k*T(n,k), k>=0) = A273720(n).

			Row 4 is 3,1,1 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 3,1,1,2,1 horizontal steps in their peaks.
Triangle starts
1;
1,1;
3,1,1;
8,3,1,1;
21,9,3,1,1

Alois P. Heinz, Rows n = 2..150, flattened
A. Blecher, C. Brennan, and A. Knopfmacher, Peaks in bargraphs, Trans. Royal Soc. South Africa, 71, No. 1, 2016, 97-103.
M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.

Cf. A000045, A088305, A082582, A273720.

eq := G = z^2*s+z*(G-z^2*s/(1-z*s)+z^2*s^2/(1-z*s))+z*G+z^2*G+z*G*(G-z^2*s/(1-z*s)+z^2/(1-z)): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 20)): for n from 2 to 16 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 2 to 16 do seq(coeff(P[n], s, j), j = 1 .. n-1) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y, t, h) option remember; expand(
      `if`(n=0, (1-t)*z^h, `if`(t<0, 0, b(n-1, y+1, 1, 0))+
      `if`(t>0 or y<2, 0, b(n, y-1, -1, 0)*z^h)+
      `if`(y<1, 0, b(n-1, y, 0, `if`(max(h, t)>0, h+1, 0)))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=1..n-1))(b(n, 0$3)):
seq(T(n), n=2..16);  # Alois P. Heinz, Jun 06 2016

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

G.f.: G(s,z), where s marks number of horizontal steps in the peaks and z marks semiperimeter, satisfies the equation given in the Maple  program.
G.f.: G(w,z), where w marks number of horizontal steps in the peaks and z marks semiperimeter, satisfies eq. (7) of the Blecher et al. reference, where one has to set x = z and y = z.
The trivariate g.f. G = G(t,s,z), where t marks number of peaks, s marks number of horizontal steps in the peaks, and z marks semiperimeter, satisfies z*(1-z)*(1-s*z)*G^2-(1-3*z-s*z+z^2+3*s*z^2-s*z^3+t*s*z^3-t*s*z^4)*G + t*s*z^2*(1-z)^2 = 0.

A277973 Sum of horizontal positions of the first peak in all bargraphs of semiperimeter n.

Original entry on oeis.org

0, 0, 0, 1, 6, 25, 91, 311, 1029, 3346, 10778, 34544, 110444, 352785, 1126885, 3601617, 11521648, 36899528, 118322448, 379908707, 1221423149, 3932113059, 12675055399, 40909511880, 132200481507, 427718677728, 1385419058692, 4492446685542, 14582927712740, 47385785436719

Offset: 1

Arnold Knopfmacher, Nov 07 2016

nonn

Horizontal position is x-coordinate of the start of the leftmost horizontal step of the first peak.

			For n = 4, a(4) = 1, as only the bargraph with first column of height one and second column of height two has horizontal position 1, all other cases are zero.

Andrew Howroyd, Table of n, a(n) for n = 1..500
A. Blecher, C. Brennan, and A. Knopfmacher, Peaks in bargraphs, Trans. Royal Soc. South Africa, 71, No. 1, 2016, 97-103.

Cf. A271941, A273720, A273345, A277999.

seq(n) = my(r=sqrt((1 - x)*(1 - 3*x - x^2 - x^3) + O(x^(n-2)))); Vec(2*x^3*(1 + x^2 - r) / ((1 - x)*(1 - 2*x - x^2 + r)^2), -n) \\ Andrew Howroyd, Jan 12 2024

G.f.: (2*x^3*(x^2-sqrt(x^4+2*x^2-4*x+1)+1)) / ((1-x)*(-x^2+sqrt(x^4+2*x^2-4*x+1)-2*x+1)^2).

A277999 Sum of distances between leftmost and rightmost peaks in all bargraphs of semiperimeter n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 9, 53, 261, 1165, 4887, 19642, 76519, 291095, 1086946, 3998430, 14530223, 52272218, 186467253, 660449671, 2325124444, 8143334776, 28393762841, 98621419068, 341403900888, 1178425064256, 4057244213071, 13937739553781, 47786215201214, 163554669548711

Offset: 1

Arnold Knopfmacher, Nov 08 2016

nonn

			a(6)=1 since the bargraph with column heights 2,1,2 has a distance of 1 between first and last peak. All other bargraphs of semiperimeter 6 have at most one peak, hence 0 difference.

A. Blecher, C. Brennan, and A. Knopfmacher, Peaks in bargraphs, Trans. Royal Soc. South Africa, 71, No. 1, 2016, 97-103.

Cf. A271941, A273720, A273345, A277973.

my(x = 'x + O('x^30)); sqx = sqrt(x^4+2*x^2-4*x+1); concat(vector(5), Vec(-(4*x^6*(3-2*x^3+3*x^4 - sqx + x^2*(4-3*sqx) + 2*x*(sqx - 4))/((x^2-3*x+1)*sqx*(-1+2*x+x^2-sqx)^3)))) \\ Michel Marcus, Feb 25 2019

G.f.: -(4*x^6*(3-2*x^3+3*x^4 - sqx + x^2*(4-3*sqx) + 2*x*(sqx - 4))/((x^2-3*x+1)*sqx*(-1+2*x+x^2-sqx)^3)) where sqx = sqrt(x^4+2*x^2-4*x+1).

A278134 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k horizontal steps in the valleys (n>=2, k>=0).

Original entry on oeis.org

1, 2, 5, 13, 34, 1, 89, 7, 1, 233, 34, 7, 1, 610, 141, 35, 7, 1, 1597, 534, 152, 36, 7, 1, 4181, 1905, 611, 163, 37, 7, 1, 10946, 6512, 2338, 689, 174, 38, 7, 1, 28657, 21557, 8641, 2787, 768, 185, 39, 7, 1, 75025, 69593, 31104, 10921, 3252, 848, 196, 40, 7, 1

Offset: 2

Emeric Deutsch, Jan 06 2017

Number of entries in rows 2,3,4,5 is 1; number of entries in row n (n>=5) is n-4.
Sum of entries in row n = A082582(n).
T(n,0) = A001519(n-1) = F(2n-3), where F(n) are the Fibonacci numbers A000045.
Sum(k*T(n,k), k>=0) = A278135(n).

			Row 6 is 34,1 because among the 35 (=A082582(6)) bargraphs of semiperimeter 6 only one has a valley; it corresponds to the composition [2,1,2] and its width is 1.
Triangle starts:
1;
2;
5;
13;
34, 1;
89, 7, 1

A. Blecher, C. Brennan, and A. Knopfmacher, Peaks in bargraphs, Trans. Royal Soc. South Africa, 71, No. 1, 2016, 97-103.

Cf. A001519, A082582, A273719, A273720, A278135

a := t*z*(1-z)^2: b := 1-3*z-t*z+z^2+3*t*z^2-t*z^4: c := z^2*(1-z)*(1-t*z): G := RootOf(a*G^2-b*G+c = 0, G): Gser := simplify(series(G, z = 0, 20)): for n from 2 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: 1; 2; 5; 13; for n from 6 to 16 do seq(coeff(P[n], t, j), j = 0 .. n-5) end do; # yields sequence in triangular form

G.f.: G(t,z), where t marks number of horizontal steps in the valleys and z marks semiperimeter, satisfies aG^2 - bG + c = 0, where a = tz(1-z)^2, b = 1 - 3z - tz + z^2 + 3t*z^2 -tz^4, c = z^2*(1-z)(1-tz).

A278135 Number of horizontal steps in the valleys of all bargraphs having semiperimeter n (n >=2).

Original entry on oeis.org

0, 0, 0, 0, 1, 9, 51, 236, 979, 3805, 14190, 51488, 183333, 644121, 2241127, 7741378, 26593899, 90971184, 310159487, 1054693058, 3578948942, 12124108632, 41015411703, 138597840864, 467913141789, 1578497031981, 5321685955902, 17931990439148, 60397664457791, 203355625940891

Offset: 2

Emeric Deutsch, Jan 06 2017

nonn

			a(6) = 1 because among the 35 (=A082582(6)) bargraphs of semiperimeter 6 only one has a valley; it corresponds to the composition [2,1,2] and its width is 1.

A. Blecher, C. Brennan, and A. Knopfmacher, Peaks in bargraphs, Trans. Royal Soc. South Africa, 71, No. 1, 2016, 97-103.

Cf. A082582, A273719, A273720, A278134

Q := sqrt((1-z)*(1-3*z-z^2-z^3)): R := 1-7*z+17*z^2-18*z^3+9*z^4-3*z^5+z^6: g := 2*z^6/(Q*(R+(1-3*z+z^2)*(1-z)^2*Q)): gser := series(g, z = 0, 35): seq(coeff(gser, z, j), j = 2 .. 33);

G.f.: g(z) = 2z^6/(Q(R + (1-3z+z^2)(1-z)^2*Q)), where Q = sqrt((1-z)(1-3z-z^2-z^3)) and R = 1 - 7z + 17z^2 - 18z^3 + 9z^4 - 3z^5 + z^6.
a(n) = Sum(k*A278134(n,k), k>=0).
Conjecture D-finite with recurrence -7*(n+1)*(n-6)*a(n) +3*(13*n^2-69*n+14)*a(n-1) +(-61*n^2+331*n-256)*a(n-2) +3*(11*n^2-59*n+68)
*a(n-3) -(n-2)*(9*n-25)*a(n-4) +(9*n^2-55*n+80)*a(n-5) -(3*n-4)*(n-5)*a(n-6) -(n-5)*(n-6)*a(n-7)=0. - R. J. Mathar, Jul 22 2022

cp's OEIS Frontend

A273719 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k horizontal steps in the peaks (n>=2, k>=1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A277973 Sum of horizontal positions of the first peak in all bargraphs of semiperimeter n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A277999 Sum of distances between leftmost and rightmost peaks in all bargraphs of semiperimeter n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A278134 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k horizontal steps in the valleys (n>=2, k>=0).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A278135 Number of horizontal steps in the valleys of all bargraphs having semiperimeter n (n >=2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula