A188464 -id:A188464 - cp's OEIS frontend

A188463 Coefficient array of the second column of the inverse of the Riordan array ((1+(r+1)x)/(1+(r+2)x+rx^2), x/(1+(r+2)x+rx^2)).

1, 3, 1, 7, 7, 1, 15, 30, 12, 1, 31, 103, 79, 18, 1, 63, 312, 387, 166, 25, 1, 127, 873, 1586, 1085, 305, 33, 1, 255, 2314, 5768, 5719, 2545, 512, 42, 1, 511, 5899, 19261, 25994, 16661, 5285, 805, 52, 1, 1023, 14604, 60337, 106009, 92008, 41881, 10038, 1204, 63, 1

Offset: 0

Paul Barry, Apr 01 2011

First column is A000225. Row sums are A128714(n+2). Diagonal sums are A188464.

			Triangle begins
1,
3, 1,
7, 7, 1,
15, 30, 12, 1,
31, 103, 79, 18, 1,
63, 312, 387, 166, 25, 1,
127, 873, 1586, 1085, 305, 33, 1,
255, 2314, 5768, 5719, 2545, 512, 42, 1,
511, 5899, 19261, 25994, 16661, 5285, 805, 52, 1

Cf. A119308.

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

A274488 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having least column-height k (n>=2, k>=1).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 8, 3, 1, 1, 22, 8, 3, 1, 1, 62, 22, 8, 3, 1, 1, 178, 62, 22, 8, 3, 1, 1, 519, 178, 62, 22, 8, 3, 1, 1, 1533, 519, 178, 62, 22, 8, 3, 1, 1, 4578, 1533, 519, 178, 62, 22, 8, 3, 1, 1, 13800, 4578, 1533, 519, 178, 62, 22, 8, 3, 1, 1, 41937, 13800, 4578, 1533, 519, 178, 62, 22, 8, 3, 1, 1

Offset: 2

Emeric Deutsch, Jul 01 2016

T(n,k) = number of bargraphs of semiperimeter n for which the width of the leftmost horizontal segment is k. A horizontal segment is a maximal sequence of adjacent horizontal steps (1,0). Example: T(4,1)=3 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 widths of their leftmost horizontal segments are 3, 1, 1, 2, 1.

Number of entries in row n is n-1.

			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, clearly, their least column-heights are 1,1,1,2,3.
Triangle starts
1;
1,1;
3,1,1;
8,3,1,1;
22,8,3,1,1

G. C. Greubel, Rows n=2..102 of triangle, 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:1609.00088 [math.CO], 2016/2018.

Sum of entries in row n = A082582(n).
T(n,1) = A188464(n-3)(n>=3).
Sum(k*T(n,k),k>=1)= A008909(n).
Cf. A273350, A274490.

G:=(1/2)*t*(1-z)*(1-2*z-z^2-sqrt((1-z)*(1-3*z-z^2-z^3)))/(z*(1-t*z)): Gser:= simplify(series(G,z=0,28)):for n from 2 to 20 do P[n]:= sort(coeff(Gser,z,n)) end do: for n from 2 to 15 do seq(coeff(P[n],t,k),k=1..n-1) end do; # yields sequence in triangular form

gf = t(1-z)((1 - 2z - z^2 - Sqrt[(1-z)(1 - 3z - z^2 - z^3)])/(2z(1 - t z)));
Rest[CoefficientList[#, t]]& /@ Drop[CoefficientList[gf + O[z]^14, z], 2] // Flatten (* Jean-François Alcover, Nov 16 2018 *)

G.f.: t(1-z)(1-2z-z^2-sqrt((1-z)(1-3z-z^2-z^3)))/(2z(1-tz)).

A274490 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n starting with k columns of length 1 (n>=2, k>=0).

Original entry on oeis.org

0, 1, 1, 0, 1, 3, 1, 0, 1, 8, 3, 1, 0, 1, 22, 8, 3, 1, 0, 1, 62, 22, 8, 3, 1, 0, 1, 178, 62, 22, 8, 3, 1, 0, 1, 519, 178, 62, 22, 8, 3, 1, 0, 1, 1533, 519, 178, 62, 22, 8, 3, 1, 0, 1, 4578, 1533, 519, 178, 62, 22, 8, 3, 1, 0, 1, 13800, 4578, 1533, 519, 178, 62, 22, 8, 3, 1, 0, 1

Offset: 2

Emeric Deutsch, Jun 25 2016

Number of entries in row n is n.
Sum of entries in row n = A082582(n).
T(n,0) = A188464(n-3) (n>=3).
Sum_{k>=0} k*T(n,k) = A105633(n-2).

			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, clearly, they start with 3, 1, 0, 0, 0 columns of length 1.
Triangle starts
0,1;
1,0,1;
3,1,0,1;
8,3,1,0,1;
22,8,3,1,0,1

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. A082582, A105633, A188464.

G := (1-3*z+z^2+2*t*z^3-z^3-(1-z)*sqrt((1-z)*(1-3*z-z^2-z^3)))/(2*z*(1-t*z)): Gser := simplify(series(G, z = 0, 22)): 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, k), k = 0 .. n-1) end do; # yields sequence in triangular form

nmax = 12;
g = (1 - 3z + z^2 + 2t z^3 - z^3 - (1-z) Sqrt[(1-z)(1 - 3z - z^2 - z^3)])/ (2z (1 - t z));
cc = CoefficientList[g + O[z]^(nmax+1), z];
T[n_, k_] := Coefficient[cc[[n+1]], t, k];
Table[T[n, k], {n, 2, nmax}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jul 25 2018 *)

G.f.: (1-3*z+z^2+2*t*z^3-z^3-(1-z)*sqrt((1-z)*(1-3*z-z^2-z^3)))/(2*z*(1-t*z)).

cp's OEIS Frontend

A188463 Coefficient array of the second column of the inverse of the Riordan array ((1+(r+1)x)/(1+(r+2)x+rx^2), x/(1+(r+2)x+rx^2)).

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A274488 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having least column-height k (n>=2, k>=1).

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A274490 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n starting with k columns of length 1 (n>=2, k>=0).

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