A266470 -id:A266470 - cp's OEIS frontend

A266464 Number of n X 2 binary arrays with rows and columns lexicographically nondecreasing and column sums nonincreasing.

1, 2, 4, 7, 12, 19, 29, 42, 59, 80, 106, 137, 174, 217, 267, 324, 389, 462, 544, 635, 736, 847, 969, 1102, 1247, 1404, 1574, 1757, 1954, 2165, 2391, 2632, 2889, 3162, 3452, 3759, 4084, 4427, 4789, 5170, 5571, 5992, 6434, 6897, 7382, 7889, 8419, 8972, 9549, 10150

Offset: 0

R. H. Hardin, Dec 29 2015

			Some solutions for n=4:
..0..0....0..0....0..1....0..0....0..1....0..0....1..1....0..1....0..0....0..1
..0..0....0..0....0..1....1..1....1..0....0..0....1..1....1..0....0..0....1..0
..0..1....0..0....1..0....1..1....1..1....1..1....1..1....1..0....0..0....1..0
..1..0....1..1....1..0....1..1....1..1....1..1....1..1....1..1....0..0....1..0

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n=1..210 from R. H. Hardin)

Column 2 of A266470.

Partial sums of A033638.

a:= proc(n) option remember;
     `if`(n<0, 0, 1+a(n-1)+floor(n^2/4))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 27 2023

a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) -a(n-5).
From Colin Barker, Mar 21 2018: (Start)
G.f.: (x^3-x+1)/((x+1)*(x-1)^4).
a(n) = (2*n^3 + 3*n^2 + 22*n + 24) / 24 for n even.
a(n) = (2*n^3 + 3*n^2 + 22*n + 21) / 24 for n odd.
(End)

a(0)=1 prepended by Alois P. Heinz, Dec 27 2023

A266465 Number of n X 3 binary arrays with rows and columns lexicographically nondecreasing and column sums nonincreasing.

Original entry on oeis.org

2, 5, 12, 29, 67, 147, 303, 590, 1090, 1922, 3253, 5311, 8400, 12918, 19377, 28425, 40873, 57722, 80196, 109776, 148240, 197703, 260666, 340063, 439318, 562401, 713894, 899055, 1123895, 1395251, 1720873, 2109508, 2570998, 3116374, 3757967

Offset: 1

R. H. Hardin, Dec 29 2015

nonn

			Some solutions for n=4:
..0..0..0....0..0..0....0..1..1....0..1..1....0..0..1....0..0..1....0..0..1
..0..0..0....0..1..1....1..0..1....1..0..1....0..1..0....1..1..0....0..1..0
..0..0..0....1..0..0....1..1..0....1..1..0....1..0..0....1..1..1....1..0..0
..0..0..0....1..1..1....1..1..0....1..1..1....1..1..0....1..1..1....1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..200

Column 3 of A266470.

Empirical: a(n) = 5*a(n-1) - 8*a(n-2) + a(n-3) + 9*a(n-4) - 6*a(n-5) - 6*a(n-7) + 9*a(n-8) + a(n-9) - 8*a(n-10) + 5*a(n-11) - a(n-12).

Empirical g.f.: x*(2 - 5*x + 3*x^2 + 7*x^3 - 5*x^4 - x^5 - 3*x^6 + 7*x^7 - 7*x^9 + 5*x^10 - x^11) / ((1 - x)^8*(1 + x)^2*(1 + x + x^2)). - Colin Barker, Jan 10 2019

A266466 Number of nX4 binary arrays with rows and columns lexicographically nondecreasing and column sums nonincreasing.

Original entry on oeis.org

2, 6, 19, 66, 232, 794, 2574, 7797, 22058, 58469, 146027, 345578, 779240, 1682359, 3492869, 6999952, 13585959, 25610002, 47004982, 84187248, 147421139, 252829908, 425319569, 702767285, 1141951244, 1826839569, 2880058656, 4478585700

Offset: 1

R. H. Hardin, Dec 29 2015

nonn

Column 4 of A266470.

			Some solutions for n=4
..0..1..1..1....0..0..0..1....0..0..0..0....0..0..1..1....0..1..1..1
..1..0..0..1....0..0..0..1....0..1..1..1....0..1..0..0....0..1..1..1
..1..1..1..0....1..1..1..0....1..0..0..0....1..0..0..0....1..0..0..0
..1..1..1..1....1..1..1..0....1..0..0..0....1..1..0..0....1..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..67

Cf. A266470.

A266467 Number of nX5 binary arrays with rows and columns lexicographically nondecreasing and column sums nonincreasing.

Original entry on oeis.org

2, 7, 28, 137, 735, 4074, 22128, 113677, 544142, 2417707, 9991874, 38585111, 139969744, 479482160, 1558677189, 4829461991, 14318767180, 40765225229, 111787710303, 296081929163, 759290568963, 1889449601320, 4571408444209

Offset: 1

R. H. Hardin, Dec 29 2015

nonn

Column 5 of A266470.

			Some solutions for n=4
..0..0..0..0..0....0..0..0..1..1....0..0..0..0..0....0..1..1..1..1
..0..0..0..0..1....0..1..1..0..0....0..0..1..1..1....1..0..0..0..0
..0..0..1..1..0....1..0..0..0..0....1..1..0..0..0....1..0..1..1..1
..1..1..0..0..0....1..0..0..0..0....1..1..1..1..1....1..1..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..32

Cf. A266470.

A266471 Number of 4 X n binary arrays with rows and columns lexicographically nondecreasing and column sums nonincreasing.

Original entry on oeis.org

5, 12, 29, 66, 137, 261, 463, 775, 1237, 1898, 2817, 4064, 5721, 7883, 10659, 14173, 18565, 23992, 30629, 38670, 48329, 59841, 73463, 89475, 108181, 129910, 155017, 183884, 216921, 254567, 297291, 345593, 400005, 461092, 529453, 605722, 690569

Offset: 1

R. H. Hardin, Dec 29 2015

nonn

			Some solutions for n=4:
..0..0..0..0....0..0..0..0....0..1..1..1....0..0..0..0....0..0..1..1
..0..0..0..0....0..0..0..0....1..0..0..0....0..0..1..1....0..1..0..0
..1..1..1..1....0..0..0..0....1..0..1..1....1..1..0..0....1..0..0..0
..1..1..1..1....1..1..1..1....1..1..0..0....1..1..1..1....1..1..1..1

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

Row 4 of A266470.

Empirical: a(n) = (1/120)*n^5 + (1/24)*n^4 + (17/24)*n^3 - (25/24)*n^2 + (257/60)*n + 1.
Conjectures from Colin Barker, Jan 10 2019: (Start)
G.f.: x*(5 - 18*x + 32*x^2 - 28*x^3 + 11*x^4 - x^5) / (1 - x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6.
(End)

A266472 Number of 5 X n binary arrays with rows and columns lexicographically nondecreasing and column sums nonincreasing.

Original entry on oeis.org

6, 19, 67, 232, 735, 2090, 5371, 12645, 27639, 56726, 110334, 204903, 365538, 629531, 1050952, 1706538, 2703140, 4187021, 6355333, 9470138, 13875377, 20017232, 28468369, 39956595, 55398509, 75938776, 102995704, 138313857, 184024492

Offset: 1

R. H. Hardin, Dec 29 2015

nonn

			Some solutions for n=4:
..0..0..1..1....0..1..1..1....0..0..0..1....0..0..0..1....0..1..1..1
..0..1..0..0....1..0..0..0....0..1..1..1....0..1..1..0....1..0..0..1
..1..0..0..0....1..0..0..1....1..0..1..0....0..1..1..0....1..0..1..0
..1..0..0..0....1..0..1..0....1..1..0..0....1..0..0..0....1..1..0..0
..1..0..0..0....1..1..0..0....1..1..1..0....1..0..0..1....1..1..1..1

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

Row 5 of A266470.

Empirical: a(n) = (1/181440)*n^9 + (1/8064)*n^8 + (29/15120)*n^7 + (13/960)*n^6 + (301/8640)*n^5 + (101/384)*n^4 - (34619/90720)*n^3 + (37531/10080)*n^2 - (4171/2520)*n + 4.
Conjectures from Colin Barker, Jan 10 2019: (Start)
G.f.: x*(6 - 41*x + 147*x^2 - 303*x^3 + 410*x^4 - 382*x^5 + 248*x^6 - 109*x^7 + 30*x^8 - 4*x^9) / (1 - x)^10.
a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n>10.
(End)

A266473 Number of 6Xn binary arrays with rows and columns lexicographically nondecreasing and column sums nonincreasing.

Original entry on oeis.org

7, 29, 147, 794, 4074, 18808, 77320, 285494, 959672, 2975483, 8605341, 23428725, 60497931, 149066593, 352233950, 801471439, 1762213254, 3755124007, 7774777259, 15675004492, 30833594755, 59276323572, 111542905766, 205731574732

Offset: 1

R. H. Hardin, Dec 29 2015

nonn

Row 6 of A266470.

			Some solutions for n=4
..0..0..1..1....0..0..0..1....0..0..0..1....0..0..1..1....0..0..0..1
..0..1..0..0....1..1..1..0....0..0..1..0....0..1..1..1....0..0..1..0
..0..1..0..1....1..1..1..0....0..0..1..0....1..0..1..1....0..1..1..0
..1..0..0..1....1..1..1..1....0..1..0..0....1..1..0..0....1..0..0..0
..1..0..1..0....1..1..1..1....1..0..0..1....1..1..0..0....1..0..0..0
..1..1..1..0....1..1..1..1....1..1..0..0....1..1..1..1....1..1..0..1

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

Cf. A266470.

Empirical: a(n) = (1/121645100408832000)*n^19 + (1/914624815104000)*n^18 + (37/533531142144000)*n^17 + (89/31384184832000)*n^16 + (1039/12553673932800)*n^15 + (116807/62768369664000)*n^14 + (3153461/94152554496000)*n^13 + (511019/1034643456000)*n^12 + (57504877/9656672256000)*n^11 + (48689987/877879296000)*n^10 + (475429693/1207084032000)*n^9 + (2471183497/1207084032000)*n^8 + (117295069721/23538138624000)*n^7 + (79279038437/3362591232000)*n^6 + (2282457077/12108096000)*n^5 - (6773798653/40864824000)*n^4 + (20107095509/9648639000)*n^3 - (10497092849/7718911200)*n^2 + (607842269/116396280)*n + 1

A266474 Number of 7Xn binary arrays with rows and columns lexicographically nondecreasing and column sums nonincreasing.

Original entry on oeis.org

8, 42, 303, 2574, 22128, 175180, 1231170, 7652503, 42460424, 212971589, 977395362, 4147719749, 16420984346, 61103700814, 215040758842, 719522481940, 2299243641848, 7043942786447, 20758083457211, 59015575393057

Offset: 1

R. H. Hardin, Dec 29 2015

nonn

Row 7 of A266470.

			Some solutions for n=4
..0..0..0..1....0..0..1..1....0..0..1..1....0..0..0..1....0..1..1..1
..0..0..0..1....0..1..1..1....0..1..0..0....0..0..1..0....1..0..1..1
..0..1..1..0....1..0..0..0....1..0..0..1....0..1..0..0....1..0..1..1
..1..0..0..1....1..1..0..0....1..0..0..1....1..0..0..0....1..0..1..1
..1..1..1..0....1..1..0..1....1..1..0..0....1..1..1..1....1..1..0..0
..1..1..1..0....1..1..1..0....1..1..1..0....1..1..1..1....1..1..0..0
..1..1..1..1....1..1..1..1....1..1..1..0....1..1..1..1....1..1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..210
R. H. Hardin, Empirical polynomial of degree 34

Cf. A266470.

Empirical polynomial of degree 34 (see link above)

A266463 Number of n X n binary arrays with rows and columns lexicographically nondecreasing and column sums nonincreasing.

Original entry on oeis.org

2, 4, 12, 66, 735, 18808, 1231170, 223521350, 119000455681, 193166954325425, 983670413713595700

Offset: 1

R. H. Hardin, Dec 29 2015

nonn

Diagonal of A266470.

			Some solutions for n=4
..0..0..0..0....0..0..1..1....0..0..0..1....0..0..0..0....0..1..1..1
..0..0..1..1....1..1..0..1....0..1..1..0....0..0..0..1....1..0..0..1
..1..1..0..0....1..1..1..0....1..0..0..0....0..0..1..0....1..0..1..0
..1..1..1..1....1..1..1..0....1..1..1..1....1..1..0..0....1..1..0..0

Cf. A266470.

A266468 Number of nX6 binary arrays with rows and columns lexicographically nondecreasing and column sums nonincreasing.

Original entry on oeis.org

2, 8, 39, 261, 2090, 18808, 175180, 1595005, 13720886, 109830369, 814603355, 5605474331, 35922351069, 215377645103, 1213816837380, 6458765687442, 32580784869691, 156385330035183, 716642920656832

Offset: 1

R. H. Hardin, Dec 29 2015

nonn

Column 6 of A266470.

			Some solutions for n=4
..0..0..0..0..0..1....0..0..0..1..1..1....0..0..0..0..0..1....0..1..1..1..1..1
..0..0..0..0..1..0....1..1..1..0..0..0....0..0..1..1..1..0....1..0..0..1..1..1
..1..1..1..1..0..0....1..1..1..0..0..0....1..1..0..0..0..0....1..0..1..0..0..0
..1..1..1..1..1..1....1..1..1..0..0..0....1..1..1..1..1..0....1..1..0..0..0..0

Cf. A266470.

cp's OEIS Frontend

A266464 Number of n X 2 binary arrays with rows and columns lexicographically nondecreasing and column sums nonincreasing.

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A266465 Number of n X 3 binary arrays with rows and columns lexicographically nondecreasing and column sums nonincreasing.

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A266466 Number of nX4 binary arrays with rows and columns lexicographically nondecreasing and column sums nonincreasing.

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A266467 Number of nX5 binary arrays with rows and columns lexicographically nondecreasing and column sums nonincreasing.

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A266471 Number of 4 X n binary arrays with rows and columns lexicographically nondecreasing and column sums nonincreasing.

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A266472 Number of 5 X n binary arrays with rows and columns lexicographically nondecreasing and column sums nonincreasing.

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A266473 Number of 6Xn binary arrays with rows and columns lexicographically nondecreasing and column sums nonincreasing.

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A266474 Number of 7Xn binary arrays with rows and columns lexicographically nondecreasing and column sums nonincreasing.

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A266463 Number of n X n binary arrays with rows and columns lexicographically nondecreasing and column sums nonincreasing.

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A266468 Number of nX6 binary arrays with rows and columns lexicographically nondecreasing and column sums nonincreasing.

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