A223770 -id:A223770 - cp's OEIS frontend

A223764 Number of n X 2 0..1 arrays with rows, columns and antidiagonals unimodal and diagonals nondecreasing.

4, 12, 28, 56, 101, 169, 267, 403, 586, 826, 1134, 1522, 2003, 2591, 3301, 4149, 5152, 6328, 7696, 9276, 11089, 13157, 15503, 18151, 21126, 24454, 28162, 32278, 36831, 41851, 47369, 53417, 60028, 67236, 75076, 83584, 92797, 102753, 113491, 125051

Offset: 1

R. H. Hardin, Mar 27 2013

nonn

Column 2 of A223770.

			Some solutions for n=3:
..0..0....0..0....0..0....1..0....1..1....0..0....0..1....0..1....0..0....0..1
..0..0....0..0....1..1....0..1....0..1....1..0....1..1....0..0....1..0....0..1
..1..1....0..1....1..1....0..0....0..1....0..1....0..1....0..0....1..1....1..0

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A223770.

Empirical: a(n) = (1/24)*n^4 + (1/4)*n^3 + (35/24)*n^2 + (5/4)*n + 1.
Conjectures from Colin Barker, Feb 21 2018: (Start)
G.f.: x*(2 - 2*x + x^2)^2 / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.
(End)

A223765 Number of n X 3 0..1 arrays with rows, columns and antidiagonals unimodal and diagonals nondecreasing.

Original entry on oeis.org

7, 28, 71, 155, 317, 607, 1097, 1887, 3112, 4950, 7631, 11447, 16763, 24029, 33793, 46715, 63582, 85324, 113031, 147971, 191609, 245627, 311945, 392743, 490484, 607938, 748207, 914751, 1111415, 1342457, 1612577, 1926947, 2291242, 2711672

Offset: 1

R. H. Hardin, Mar 27 2013

nonn

Column 3 of A223770.

			Some solutions for n=3:
..0..0..0....0..0..0....0..0..0....0..0..0....0..0..1....1..1..0....1..1..0
..0..0..0....1..0..0....0..0..1....0..1..1....0..1..1....0..1..1....1..1..1
..1..0..0....1..1..1....1..1..0....0..0..1....1..1..1....0..1..1....1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A223770.

Empirical: a(n) = (1/720)*n^6 + (1/240)*n^5 + (41/144)*n^4 - (13/48)*n^3 + (2237/360)*n^2 - (217/30)*n + 19 for n>2.
Conjectures from Colin Barker, Aug 22 2018: (Start)
G.f.: x*(7 - 21*x + 22*x^2 + x^3 - 12*x^4 - 9*x^5 + 26*x^6 - 17*x^7 + 4*x^8) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>9.
(End)

A223766 Number of n X 4 0..1 arrays with rows, columns and antidiagonals unimodal and diagonals nondecreasing.

Original entry on oeis.org

11, 56, 155, 361, 782, 1601, 3141, 5907, 10678, 18618, 31422, 51505, 82243, 128276, 195884, 293448, 432009, 625939, 893739, 1258980, 1751404, 2408203, 3275495, 4410017, 5881056, 7772640, 10186012, 13242411, 17086185, 21888262, 27850006

Offset: 1

R. H. Hardin, Mar 27 2013

nonn

Column 4 of A223770.

			Some solutions for n=3:
..1..1..1..0....0..0..1..1....0..0..1..0....1..0..0..0....0..1..1..0
..0..1..1..1....0..1..1..1....1..1..1..1....0..1..0..0....0..1..1..1
..0..1..1..1....1..1..1..1....1..1..1..1....0..0..1..1....0..0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A223770.

Empirical: a(n) = (1/40320)*n^8 - (1/10080)*n^7 + (3/320)*n^6 + (1/45)*n^5 + (1109/1920)*n^4 - (4837/1440)*n^3 + (422483/10080)*n^2 - (109337/840)*n + 226 for n>4.
Conjectures from Colin Barker, Aug 22 2018: (Start)
G.f.: x*(11 - 43*x + 47*x^2 + 58*x^3 - 205*x^4 + 209*x^5 - 42*x^6 - 150*x^7 + 256*x^8 - 245*x^9 + 151*x^10 - 55*x^11 + 9*x^12) / (1 - x)^9.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>13.
(End)

A223767 Number of nX5 0..1 arrays with rows, columns and antidiagonals unimodal and diagonals nondecreasing.

Original entry on oeis.org

16, 101, 317, 782, 1748, 3699, 7564, 14980, 28778, 53619, 96979, 170525, 292045, 488120, 797778, 1277432, 2007477, 3101006, 4715203, 7066083, 10447376, 15254495, 22014688, 31424652, 44397084, 62117861, 86115779, 118347041, 161296967

Offset: 1

R. H. Hardin Mar 27 2013

nonn

Column 5 of A223770

			Some solutions for n=3
..0..1..0..0..0....0..0..1..1..0....0..0..0..0..0....0..0..0..0..0
..0..1..1..0..0....0..0..1..1..1....0..1..1..0..0....0..0..0..0..1
..0..1..1..1..1....0..0..0..1..1....0..0..1..1..0....0..0..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = (1/3628800)*n^10 - (1/241920)*n^9 + (23/120960)*n^8 - (43/40320)*n^7 + (11293/172800)*n^6 - (1721/2304)*n^5 + (1557823/181440)*n^4 - (3058823/60480)*n^3 + (99347/350)*n^2 - (196177/210)*n + 1664 for n>6

A223768 Number of nX6 0..1 arrays with rows, columns and antidiagonals unimodal and diagonals nondecreasing.

Original entry on oeis.org

22, 169, 607, 1601, 3699, 8007, 16742, 34096, 67890, 132115, 251110, 465910, 843960, 1493505, 2584666, 4379914, 7277630, 11873716, 19047873, 30083272, 46831011, 71934087, 109129744, 163654129, 242779358, 356520541, 518559236, 747440414

Offset: 1

R. H. Hardin Mar 27 2013

nonn

Column 6 of A223770

			Some solutions for n=3
..0..1..0..0..0..0....1..0..0..0..0..0....0..1..0..0..0..0....0..1..1..1..1..0
..1..1..1..1..0..0....1..1..1..0..0..0....0..1..1..1..0..0....0..1..1..1..1..1
..0..1..1..1..1..1....0..1..1..1..1..0....1..1..1..1..1..1....0..0..1..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = (1/479001600)*n^12 - (1/15966720)*n^11 + (23/8709120)*n^10 - (59/1451520)*n^9 + (22591/14515200)*n^8 - (1159/53760)*n^7 + (713737/1741824)*n^6 - (1665749/290304)*n^5 + (795178259/10886400)*n^4 - (220668799/362880)*n^3 + (73774259/20790)*n^2 - (340682581/27720)*n + 20228 for n>8

A223769 Number of nX7 0..1 arrays with rows, columns and antidiagonals unimodal and diagonals nondecreasing.

Original entry on oeis.org

29, 267, 1097, 3141, 7564, 16742, 35513, 73427, 149053, 297518, 583860, 1125118, 2126567, 3939287, 7150127, 12719513, 22189580, 37994506, 63918056, 105761409, 172308985, 276713314, 438464653, 686170386, 1061447357, 1624332246

Offset: 1

R. H. Hardin Mar 27 2013

nonn

Column 7 of A223770

			Some solutions for n=3
..1..0..0..0..0..0..0....0..1..1..0..0..0..0....0..1..1..0..0..0..0
..1..1..1..0..0..0..0....0..1..1..1..0..0..0....0..1..1..1..0..0..0
..1..1..1..1..1..0..0....0..0..1..1..1..0..0....1..1..1..1..1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = (1/87178291200)*n^14 - (1/1779148800)*n^13 + (1/38320128)*n^12 - (127/191600640)*n^11 + (311/12441600)*n^10 - (2021/3225600)*n^9 + (2304587/121927680)*n^8 - (6921119/17418240)*n^7 + (278861369/43545600)*n^6 - (1599083081/21772800)*n^5 + (2209531079/3421440)*n^4 - (4215012997/997920)*n^3 + (3157078175059/151351200)*n^2 - (25129653953/360360)*n + 120027 for n>10

A223763 Number of n X n 0..1 arrays with rows, columns and antidiagonals unimodal and diagonals nondecreasing.

Original entry on oeis.org

2, 12, 71, 361, 1748, 8007, 35513, 153450, 651075, 2723829, 11275932, 46302339, 188958191, 767474182, 3105857015

Offset: 1

R. H. Hardin Mar 27 2013

nonn

Diagonal of A223770

			Some solutions for n=3
..0..0..0....0..0..0....1..1..1....1..0..0....0..0..0....1..1..1....0..0..1
..0..0..0....0..1..1....1..1..1....0..1..0....0..1..1....0..1..1....0..0..0
..0..0..1....0..1..1....0..1..1....0..1..1....0..0..1....0..1..1....0..0..0

cp's OEIS Frontend

A223764 Number of n X 2 0..1 arrays with rows, columns and antidiagonals unimodal and diagonals nondecreasing.

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A223765 Number of n X 3 0..1 arrays with rows, columns and antidiagonals unimodal and diagonals nondecreasing.

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A223766 Number of n X 4 0..1 arrays with rows, columns and antidiagonals unimodal and diagonals nondecreasing.

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A223767 Number of nX5 0..1 arrays with rows, columns and antidiagonals unimodal and diagonals nondecreasing.

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A223768 Number of nX6 0..1 arrays with rows, columns and antidiagonals unimodal and diagonals nondecreasing.

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A223769 Number of nX7 0..1 arrays with rows, columns and antidiagonals unimodal and diagonals nondecreasing.

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A223763 Number of n X n 0..1 arrays with rows, columns and antidiagonals unimodal and diagonals nondecreasing.

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