A208355 -id:A208355 - cp's OEIS frontend

A370060 Array read by antidiagonals: T(n,k) is the number of achiral dissections of a polygon into n k-gons by nonintersecting diagonals rooted at a cell, n >= 1, k >= 3.

1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 2, 4, 2, 1, 1, 4, 4, 12, 5, 1, 1, 3, 6, 9, 18, 5, 1, 1, 5, 6, 26, 22, 55, 14, 1, 1, 4, 8, 21, 45, 52, 88, 14, 1, 1, 6, 8, 45, 51, 204, 140, 273, 42, 1, 1, 5, 10, 38, 84, 190, 380, 340, 455, 42, 1, 1, 7, 10, 69, 92, 500, 506, 1771, 969, 1428, 132

Offset: 1

Andrew Howroyd, Feb 08 2024

The polygon prior to dissection will have n*(k-2)+2 sides.

			Array begins:
=============================================
n\k|  3   4   5    6    7    8    9    10 ...
---+-----------------------------------------
1  |  1   1   1    1    1    1    1     1 ...
2  |  1   1   1    1    1    1    1     1 ...
3  |  1   3   2    4    3    5    4     6 ...
4  |  2   4   4    6    6    8    8    10 ...
5  |  2  12   9   26   21   45   38    69 ...
6  |  5  18  22   45   51   84   92   135 ...
7  |  5  55  52  204  190  500  468   992 ...
8  | 14  88 140  380  506 1008 1240  2100 ...
9  | 14 273 340 1771 1950 6200 6545 15990 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.
Wikipedia, Fuss-Catalan number

Columns k=3..6 are A208355(n-1), A124817(n-1), A369472, A370061.

Cf. A070914 (rooted), A295222 (oriented), A295259 (unoriented), A369929, A370062 (achiral unrooted).

\\ here u is Fuss-Catalan sequence with p = k-1.
u(n, k, r) = {r*binomial((k - 1)*n + r, n)/((k - 1)*n + r)}
T(n, k) = {if(k%2, if(n%2, u((n-1)/2, k, (k-1)/2), u(n/2-1, k, (k-1))), if(n%2, u((n-1)/2, k, k/2+1), u(n/2-1, k, k)) )}
for(n=1, 9, for(k=3, 10, print1(T(n, k), ", ")); print);

T(n,k) = 2*A295259(n,k) - A295222(n,k).

T(n,2*k+1) = A370062(n,2*k+1).

A208101 Triangle read by rows: T(n,0) = 1; for n > 0: T(n,1) = n, for n>1: T(n,n) = T(n-1,n-2); T(n,k) = T(n-2,k-1) + T(n-1,k) for k: 1 < k < n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 2, 1, 4, 3, 5, 2, 1, 5, 4, 9, 5, 5, 1, 6, 5, 14, 9, 14, 5, 1, 7, 6, 20, 14, 28, 14, 14, 1, 8, 7, 27, 20, 48, 28, 42, 14, 1, 9, 8, 35, 27, 75, 48, 90, 42, 42, 1, 10, 9, 44, 35, 110, 75, 165, 90, 132, 42, 1, 11, 10, 54, 44, 154, 110

Offset: 0

Reinhard Zumkeller, Mar 04 2012

Another variant of Pascal's triangle, cf. A007318.

			The triangle begins:
0:                    1
1:                  1   1
2:                1   2   1
3:              1   3   2   2
4:            1   4   3   5   2
5:          1   5   4   9   5   5
6:        1   6   5  14   9  14   5
7:      1   7   6  20  14  28  14  14
8:    1   8   7  27  20  48  28  42  14
9:  1   9   8  35  27  75  48  90  42  42

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A208976 (row sums), A101461 (row max), A208983 (central), A208355 (right edge), A074909.

a208101 n k = a208101_tabl !! n !! k
a208101_row n = a208101_tabl !! n
a208101_tabl =  iterate
   (\row -> zipWith (+) ([0,1] ++ init row) (row ++ [0])) [1]

T[, 0] = 1; T[n, 1] := n; T[n_, n_] := T[n-1, n-2]; T[n_, k_] /; 1Jean-François Alcover, Feb 03 2018 *)

A296664 Table read by rows, diagonals of powers of Toeplitz matrices generated by the characteristic function of 1, T(n, k) for n >= 0 and 0 <= k <= 2*floor(n/2).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 2, 2, 5, 6, 5, 2, 5, 9, 10, 9, 5, 5, 14, 19, 20, 19, 14, 5, 14, 28, 34, 35, 34, 28, 14, 14, 42, 62, 69, 70, 69, 62, 42, 14, 42, 90, 117, 125, 126, 125, 117, 90, 42, 42, 132, 207, 242, 251, 252, 251, 242, 207, 132, 42

Offset: 0

Peter Luschny, Dec 19 2017

Let v be the characteristic function of 1 (A063524) and M(n) for n >= 0 the symmetric Toeplitz matrix generated by the initial segment of v, then row n is the main diagonal of M(n)^n if n is even or the diagonal next to the main diagonal if n is odd. Note that the antidiagonals of M(n)^n are the rows of Pascal's triangle A007318.

			The first few matrices M(n)^n are:
n=0   n=1     n=2       n=3         n=4
|1|  |0 1|  |1 0 1|  |0 2 0 1|  |2 0 3 0 1|
     |1 0|  |0 2 0|  |2 0 3 0|  |0 5 0 4 0|
            |1 0 1|  |0 3 0 2|  |3 0 6 0 3|
                     |1 0 2 0|  |0 4 0 5 0|
                                |1 0 3 0 2|
The triangle starts:
0: [ 1]
1: [ 1]
2: [ 1,  2,   1]
3: [ 2,  3,   2]
4: [ 2,  5,   6,   5,   2]
5: [ 5,  9,  10,   9,   5]
6: [ 5, 14,  19,  20,  19,  14,  5]
7: [14, 28,  34,  35,  34,  28,  14]
8: [14, 42,  62,  69,  70,  69,  62, 42, 14]
9: [42, 90, 117, 125, 126, 125, 117, 90, 42]

Cf. A000108, A001405, A208355, A296663 (row sums), A296662 (odd rows), A296666 (even rows).

v := n -> `if`(n=1, 1, 0):
M := n -> LinearAlgebra:-ToeplitzMatrix([seq(v(j), j=0..n)], symmetric):
seq(convert(ArrayTools:-Diagonal(M(n)^n, n mod 2), list), n=0..10);

v[n_] := If[n == 1, 1, 0];
m[n_] := MatrixPower[ToeplitzMatrix[Table[v[k], {k, 0, n}]], n];
d[n_] := If[n == 0, {1}, Diagonal[m[n], Mod[n, 2]]];
Table[d[n], {n, 0, 10}] // Flatten

def T(n, k):
    h, e = n//2, n%2 == 0
    a = binomial(n, h) if e else binomial(2*h+1, h+1)
    if k > h:
        b = binomial(n, k-h-1) if e else binomial(2*h+1, k-h-1)
    else:
        b = binomial(n, h+k+1) if e else binomial(2*h+1, h-k-1)
    return a - b
for n in (0..9): print([T(n, k) for k in (0..2*(n//2))])

T(n, 0) = T(n, 2*floor(n/2)) = A208355(n) = A000108(floor((n+1)/2)).
T(n, floor(n/2)) = A001405(n).
Further formulas can be found in A296662 and A296666 for the cases n odd and n even.

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A370060 Array read by antidiagonals: T(n,k) is the number of achiral dissections of a polygon into n k-gons by nonintersecting diagonals rooted at a cell, n >= 1, k >= 3.

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A208101 Triangle read by rows: T(n,0) = 1; for n > 0: T(n,1) = n, for n>1: T(n,n) = T(n-1,n-2); T(n,k) = T(n-2,k-1) + T(n-1,k) for k: 1 < k < n.

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A296664 Table read by rows, diagonals of powers of Toeplitz matrices generated by the characteristic function of 1, T(n, k) for n >= 0 and 0 <= k <= 2*floor(n/2).

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