A382923
Square array A(n,k), n >= 0, k >= 0, read by downward antidiagonals: A(n,k) is the number of m-compositions of n with k zeros.
Original entry on oeis.org
1, 0, 1, 0, 2, 3, 0, 3, 5, 7, 0, 4, 13, 16, 16, 0, 5, 14, 33, 40, 35, 0, 6, 29, 70, 105, 100, 75, 0, 7, 27, 88, 207, 292, 244, 159, 0, 8, 51, 152, 336, 604, 758, 576, 334, 0, 9, 44, 206, 588, 1161, 1749, 1920, 1329, 696, 0, 10, 79, 300, 882, 2076, 3685, 4924, 4802, 3028, 1442
Offset: 0
Square array begins:
1, 0, 0, 0, 0, 0, ...
1, 2, 3, 4, 5, 6, ...
3, 5, 13, 14, 29, 27, ...
7, 16, 33, 70, 88, 152, ...
16, 40, 105, 207, 336, 588, ...
35, 100, 292, 604, 1161, 2076, ...
...
A(2,0) = 3 counts:
[2], [1,1], [1]
[1].
A(2,1) = 5 counts:
[2] [0] [1] [1] [0]
[0], [2], [1] [0] [1]
[0], [1], [1].
-
G_tx(max_row) = {my(row = max_row, N = row*2, m = List([concat([1],vector(row-1,i,0))]), x='x+O('x^N), h=1 + sum(m=1,N,-1+ 1/(1 + t^m - (t + x/(1-x))^m))); for(n=1,row, listput(m,Vecrev(polcoeff(h, n))[1..row])); matrix(row, row, i,j, m[i][j])}
G_tx(10)
A181294
Number of 0's in all 2-compositions of n. A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.
Original entry on oeis.org
0, 2, 10, 46, 198, 816, 3264, 12776, 49192, 186976, 703328, 2623072, 9712864, 35746816, 130873088, 476961920, 1731331200, 6262393344, 22580421120, 81188953600, 291176175104, 1041867493376, 3720118018048, 13257657264128
Offset: 0
a(2)=10 because the 2-compositions of 2, written as (top row / bottom row), are (1/1),(0/2),(2/0),(1,0/0,1),(0,1/1,0),(1,1/0,0),(0,0/1,1), having 0 + 1 + 1 + 2 + 2 + 2 + 2 = 10 zeros.
- G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European Journal of Combinatorics, 28, 2007, 1724-1741.
-
g := 2*z*(1-z)^3/(1-4*z+2*z^2)^2: gser := series(g, z = 0, 30): seq(coeff(gser, z, n), n = 0 .. 25);
A181330
Triangle read by rows: T(n,k) is the number of 2-compositions of n having k 0's in the top row A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.
Original entry on oeis.org
1, 1, 1, 3, 3, 1, 8, 10, 5, 1, 21, 32, 21, 7, 1, 55, 99, 80, 36, 9, 1, 144, 299, 286, 160, 55, 11, 1, 377, 887, 978, 650, 280, 78, 13, 1, 987, 2595, 3236, 2482, 1275, 448, 105, 15, 1, 2584, 7508, 10438, 9054, 5377, 2261, 672, 136, 17, 1, 6765, 21526, 32991, 31882
Offset: 0
T(2,1)=3 because we have (0/2), (1,0/0,1), and (0,1/1,0) (the 2-compositions are written as (top row / bottom row)).
Triangle starts:
1;
1,1;
3,3,1;
8,10,5,1;
21,32,21,7,1;
55,99,80,36,9,1;
-
G := (1-z)^2/(1-3*z+z^2-t*z*(1-z)): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 10 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 10 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form
A382924
Number of m-compositions of n with n zeros.
Original entry on oeis.org
1, 2, 13, 70, 336, 2076, 11091, 65210, 365661, 2159354, 11713047, 71427504, 392916687, 2245186352, 13527678851, 73679458270, 429472428457, 2553994191220, 14264421153074, 80483620074092, 489077890675807, 2768919905996888, 15394229582049408, 91794448088043258
Offset: 0
a(2) = 13 counts:
[2] [0] [0] [1] [1] [1] [0] [0] [0] [1][1] [1][0] [0][0] [0][1]
[0] [2] [0] [1] [0] [0] [1] [1] [0] [0][0], [0][1], [1][1], [1][0].
[0], [0], [2], [0] [1] [0] [1] [0] [1]
[0], [0], [1], [0], [1], [1],
-
G_tx(max_row) = {my(row = max_row, N = row*2, m = List([concat([1],vector(row-1,i,0))]), x='x+O('x^N), h=1 + sum(m=1,N,-1+ 1/(1 + t^m - (t + x/(1-x))^m))); for(n=1,row, listput(m,Vecrev(polcoeff(h, n))[1..row])); matrix(row, row, i,j, m[i][j])}
A382924(max_n) ={my(A=G_tx(max_n)); vector(max_n,i,A[i,i])}
A382924(20)
Showing 1-4 of 4 results.
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