A273903 -id:A273903 - cp's OEIS frontend

A273902 Number of odd-length columns in all bargraphs having semiperimeter n (n>=2).

1, 2, 6, 20, 64, 204, 656, 2120, 6873, 22350, 72881, 238232, 780384, 2561164, 8419766, 27721784, 91397927, 301710074, 997087170, 3298556716, 10922576840, 36199599880, 120068987717, 398547827336, 1323821438203, 4400043488826, 14633372199291

Offset: 2

Emeric Deutsch, Jun 22 2016

nonn

			a(4) = 6 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1],[1,2],[2,1],[2,2],[3] and, clearly, they have 3,1,1,0,1 columns of odd length.

Alois P. Heinz, Table of n, a(n) for n = 2..1000
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. A082582, A273901, A273903, A273904.

Q := sqrt((1-z)*(1-3*z-z^2-z^3)): g := (((1-z)*(1-z-z^2-z^3)-(1-z^2)*Q)*(1/2))/((1+z^2)*Q): gser := series(g, z = 0, 40): seq(coeff(gser, z, m), m = 2 .. 35);
# second Maple program:
a:= proc(n) option remember; `if`(n<7, [0$2, 1, 2, 6, 20, 64]
       [n+1], ((n-1)*(55*n-178)*a(n-1)-(2*(n-2))*(32*n-143)*
       a(n-2)+(501-370*n+69*n^2)*a(n-3)-(524-443*n+64*n^2)*
       a(n-4)+(526-215*n+21*n^2)*a(n-5)-(4*(3*n+2))*(n-6)*
       a(n-6)+(n-7)*(-29+7*n)*a(n-7))/ (n*(12*n-35)))
    end:
seq(a(n), n=2..35);  # Alois P. Heinz, Jun 23 2016

Q = Sqrt[(1-z)*(1-3*z-z^2-z^3)]; g = (((1-z)*(1-z-z^2-z^3) - (1-z^2)*Q)*(1/2))/((1+z^2)*Q); gser = g + O[z]^40; CoefficientList[gser, z][[3 ;; -1]] (* Jean-François Alcover, Oct 04 2016, adapted from Maple *)

G.f.: g(z) = ((1-z)(1-z-z^2-z^3)-(1-z^2)Q)/(2(1+z^2)*Q), where Q = sqrt((1-z)(1-3z-z^2-z^3)).
a(n) = Sum(k*A273901(n,k), k>=0).
D-finite with recurrence n*(12*n-35)*a(n) -(n-1)*(55*n-178)*a(n-1) +2*(n-2)*(32*n-143)*a(n-2) +(-69*n^2+370*n-501)*a(n-3) +(64*n^2-443*n+524)*a(n-4) +(-21*n^2+215*n-526)*a(n-5) +4*(3*n+2)*(n-6)*a(n-6) -(7*n-29)*(n-7)*a(n-7)=0. - R. J. Mathar, Jul 26 2022

A273904 Number of even-length columns in all bargraphs having semiperimeter n (n>=2).

Original entry on oeis.org

0, 1, 4, 13, 44, 149, 498, 1656, 5498, 18236, 60456, 200409, 664464, 2203755, 7311894, 24271290, 80605250, 267821525, 890305418, 2961015981, 9852481830, 32798011430, 109229396466, 363927233758, 1213012655490, 4044684629394, 13491663770344

Offset: 2

Emeric Deutsch, Jun 23 2016

nonn

			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, clearly, they have 0,1,1,2,0 columns of even length.

Alois P. Heinz, Table of n, a(n) for n = 2..1000
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. A082582, A273901, A273902, A273903.

Q := sqrt((1-z)*(1-3*z-z^2-z^3)): g := (((1-z)*(1-3*z+z^2-z^3)-(1-z)^2*Q)*(1/2))/(z*(1+z^2)*Q): gser := series(g, z = 0, 40): seq(coeff(gser, z, m), m = 2 .. 35);
# second Maple program:
a:= proc(n) option remember; `if`(n<7, [0$3, 1, 4, 13, 44]
      [n+1], ((7*n-22)*(n-6)*a(n-7) -(5*n^2-21*n+6)*a(n-6)+
      (21*n^2-180*n+404)*a(n-5) -(43*n^2-265*n+332)*a(n-4)
      +(41*n^2-226*n+308)*a(n-3) -(43*n^2-257*n+308)*a(n-2)
      +(27*n^2-110*n+36)*a(n-1))/ ((n+1)*(5*n-18)))
    end:
seq(a(n), n=2..40);  # Alois P. Heinz, Jun 24 2016

Q = Sqrt[(1-z)*(1-3*z-z^2-z^3)]; g = (((1-z)*(1-3*z+z^2-z^3) - (1-z)^2 * Q)*(1/2))/(z*(1+z^2)*Q); gser = g + O[z]^40; CoefficientList[gser, z][[3 ;; -1]] (* Jean-François Alcover, Oct 04 2016, adapted from Maple *)

G.f.: g(z)=((1-z)(1-3z+z^2-z^3)-(1-z)^2*Q)/(2z(1+z^2)*Q), where Q = sqrt((1-z)(1-3z-z^2-z^3)).

a(n) = Sum(k*A273903(n,k), k>=0).

A273901 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k odd-length columns (n>=2, k>=0).

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 3, 0, 1, 2, 4, 6, 0, 1, 4, 11, 9, 10, 0, 1, 8, 24, 33, 16, 15, 0, 1, 16, 56, 80, 76, 25, 21, 0, 1, 33, 128, 218, 200, 150, 36, 28, 0, 1, 69, 297, 558, 630, 420, 267, 49, 36, 0, 1, 146, 688, 1445, 1776, 1515, 784, 441, 64, 45, 0, 1, 312, 1601, 3684, 5091, 4635, 3213, 1344, 688, 81, 55, 0, 1

Offset: 2

Emeric Deutsch, Jun 22 2016

Number of entries in row n is n.
Sum of entries in row n = A082582(n).
T(n,0) = A004149(n-1).
Sum(k*T(n,k), k>=0) = A273902(n).

			Row 4 is 1,3,0,1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] and, clearly, they have 3, 1, 1, 0, 1 columns of odd length.
Triangle starts
0,1;
1,0,1;
1,3,0,1;
2,4,6,0,1;
4,11,9,10,0,1

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. A004149, A082582, A273902, A273903, A273904.

eq := z*(1+t*z^2)*f^2-(1-z-t*z+t*z^2-t*z^4-z^3-z^2-t*z^3)*f+z^2*(t*z^2+z-t*z+t) = 0: f:= RootOf(eq,f): fser:=simplify(series(f, z = 0,15)): for n from 2 to 12 do P[n] := sort(coeff(fser, z, n)) end do: for n from 2 to 12 do seq(coeff(P[n],t,k),k=0..n-1)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>0 or y<2, 0,
       b(n, y-1, -1))+ `if`(y<1, 0, b(n-1, y, 0)*z^
      `if`(y::odd, 1, 0))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..n-1))(b(n, 0$2)):
seq(lprint(T(n)), n=2..15);  # Alois P. Heinz, Jun 24 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>0 || y<2, 0, b[n, y-1, -1]] + If[y<1, 0, b[n-1, y, 0]*z^If[OddQ[y], 1, 0]]]]; T[n_] := Function [p, Table[Coefficient[p, z, i], {i, 0, n-1}]][b[n, 0, 0]]; Table[T[n], {n, 2, 15}] // Flatten (* Jean-François Alcover, Nov 29 2016 after Alois P. Heinz *)

G.f.: G = G(t,z) satisfies aG^2 + bG + c = 0, where a = z(1+tz^2), b = tz^3 + z^2 + z^3 + tz^4 - tz^2 + tz + z - 1, c = z^2*(tz^2 + z - tz + t).

The trivariate g.f. G(t,s,z), where t (s) marks number of odd-length (even-length) columns and z marks semiperimeter, satisfies AG^2 + BG + C = 0, where A = z(tsz^2 - tsz + tz +s), B = tsz^4+(t+s)z^3+(1-ts)z^2+(t+s)z-1, C = tsz^4+s(1-t)z^3+tz^2.

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A273902 Number of odd-length columns in all bargraphs having semiperimeter n (n>=2).

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A273904 Number of even-length columns in all bargraphs having semiperimeter n (n>=2).

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A273901 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k odd-length columns (n>=2, k>=0).

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