A223838 -id:A223838 - cp's OEIS frontend

A224173 T(n,k) = number of n X k 0..3 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

4, 16, 10, 50, 100, 20, 130, 684, 400, 35, 296, 3526, 4739, 1225, 56, 610, 14751, 38561, 22988, 3136, 84, 1163, 52591, 242114, 272130, 87878, 7056, 120, 2083, 165212, 1253770, 2335459, 1460836, 282372, 14400, 165, 3544, 468292, 5588411, 15925611

Offset: 1

R. H. Hardin, Mar 31 2013

			Table starts:
    4    16       50       130         296          610           1163
   10   100      684      3526       14751        52591         165212
   20   400     4739     38561      242114      1253770        5588411
   35  1225    22988    272130     2335459     15925611       91494280
   56  3136    87878   1460836    16625026    143558572     1012166273
   84  7056   282372   6425876    95808564   1038484760     8857798353
  120 14400   794220  24197608   468021427   6360047093    65713691148
  165 27225  2010035  80350989  1994287334  33901838632   426013124302
  220 48400  4668304 240416852  7568051210 160168789130  2451904991177
  286 81796 10095924 658890738 25994968917 680269560125 12667946702827
  ...
Some solutions for n=3 k=4
..0..0..1..0....0..0..1..2....0..0..3..0....0..2..0..0....0..3..3..1
..1..3..3..1....0..1..3..2....3..3..3..1....1..2..0..0....1..3..3..1
..1..3..3..3....0..3..3..2....3..3..3..2....2..2..1..0....1..3..3..3

R. H. Hardin, Table of n, a(n) for n = 1..178

Main diagonal is A224167.
Columns 1..7 are A000292(n+1), A001249, A224168, A224169, A224170, A224171, A224172.
Rows 1..7 are A223659, A223865, A224174, A224175, A224176, A224177, A224178.
Cf. A223838.

Empirical: columns k=1..7 are polynomials of degree 3*k for n>0,0,0,3,6,9,12.

Empirical: rows n=1..5 are polynomials of degree 6*n for k>0,0,0,2,6.

Name corrected by Andrew Howroyd, Mar 18 2025

A224262 T(n,k) = number of n X k 0..2 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

Original entry on oeis.org

3, 9, 6, 22, 36, 10, 46, 158, 100, 15, 86, 548, 648, 225, 21, 148, 1600, 3096, 2017, 441, 28, 239, 4102, 12032, 12467, 5246, 784, 36, 367, 9503, 40182, 59855, 41012, 11990, 1296, 45, 541, 20299, 119367, 240829, 238366, 116692, 24842, 2025, 55, 771, 40570

Offset: 1

R. H. Hardin, Apr 02 2013

			Table starts:
   3    9     22      46       86       148        239         367          541
   6   36    158     548     1600      4102       9503       20299        40570
  10  100    648    3096    12032     40182     119367      322885       808618
  15  225   2017   12467    59855    240829     850875     2717731      8000608
  21  441   5246   41012   238366   1122522    4542734    16423026     54399996
  28  784  11990  116692   816361   4480391   20568693    82733667    301228048
  36 1296  24842  296646  2485967  15921905   83124099   371699763   1478187738
  45 2025  47643  688533  6868203  51343083  306179180  1530419762   6671184875
  55 3025  85838 1482310 17467782 152072846 1038489172  5835731860  28072690614
  66 4356 146878 2995516 41364960 417672794 3266157979 20709405119 110622071553
  ...
Some solutions for n=3 k=4:
..1..1..0..0....1..1..1..1....1..1..2..1....0..2..1..0....0..0..0..0
..2..1..1..0....1..2..2..1....2..2..2..1....0..2..1..1....0..0..2..0
..2..2..1..1....2..2..2..1....2..2..2..2....0..2..2..1....0..0..2..2

R. H. Hardin, Table of n, a(n) for n = 1..241

Main diagonal is A224256.
Columns 1..7 are A000217(n+1), A000537(n+1), A224257, A224258, A224259, A224260, A224261.
Rows 1..7 are A223718, A223919, A224263, A224264, A224265, A224266, A224267.
Cf. A223838.

Empirical: columns k=1..7 are polynomials of order 2*k for n>0,0,0,2,4,6,8.

Empirical: rows n=1..7 are polynomials of degree 4*n for k>0,0,0,2,4,6,8.

Name corrected by Andrew Howroyd, Mar 18 2025

A223833 Number of n X 3 0..1 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

Original entry on oeis.org

7, 22, 48, 89, 149, 232, 342, 483, 659, 874, 1132, 1437, 1793, 2204, 2674, 3207, 3807, 4478, 5224, 6049, 6957, 7952, 9038, 10219, 11499, 12882, 14372, 15973, 17689, 19524, 21482, 23567, 25783, 28134, 30624, 33257, 36037, 38968, 42054, 45299, 48707, 52282

Offset: 1

R. H. Hardin, Mar 27 2013

nonn

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

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

Column 3 of A223838.

Empirical: a(n) = (2/3)*n^3 + (3/2)*n^2 + (35/6)*n - 1.
Conjectures from Colin Barker, Aug 23 2018: (Start)
G.f.: x*(7 - 6*x + 2*x^2 + x^3) / (1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.
(End)

Name corrected by Andrew Howroyd, Mar 18 2025

A223834 Number of n X 4 0..1 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

Original entry on oeis.org

11, 46, 118, 249, 471, 824, 1356, 2123, 3189, 4626, 6514, 8941, 12003, 15804, 20456, 26079, 32801, 40758, 50094, 60961, 73519, 87936, 104388, 123059, 144141, 167834, 194346, 223893, 256699, 292996, 333024, 377031, 425273, 478014, 535526, 598089

Offset: 1

R. H. Hardin, Mar 27 2013

nonn

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

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

Column 4 of A223838.

Empirical: a(n) = (1/3)*n^4 + (2/3)*n^3 + (31/6)*n^2 + (71/6)*n - 9 for n>1.
Conjectures from Colin Barker, Aug 23 2018: (Start)
G.f.: x*(11 - 9*x - 2*x^2 + 9*x^3 + x^4 - 2*x^5) / (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>6.
(End)

Name corrected by Andrew Howroyd, Mar 18 2025

A223835 Number of n X 5 0..1 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

Original entry on oeis.org

16, 86, 255, 596, 1240, 2388, 4325, 7436, 12222, 19316, 29499, 43716, 63092, 88948, 122817, 166460, 221882, 291348, 377399, 482868, 610896, 764948, 948829, 1166700, 1423094, 1722932, 2071539, 2474660, 2938476, 3469620, 4075193, 4762780, 5540466

Offset: 1

R. H. Hardin, Mar 27 2013

nonn

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

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

Column 5 of A223838.

Empirical: a(n) = (2/15)*n^5 + (1/6)*n^4 + (19/6)*n^3 + (28/3)*n^2 + (126/5)*n - 36 for n>2.
Conjectures from Colin Barker, Aug 23 2018: (Start)
G.f.: x*(16 - 10*x - 21*x^2 + 36*x^3 + 9*x^4 - 18*x^5 + 2*x^6 + 2*x^7) / (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>8.
(End)

Name corrected by Andrew Howroyd, Mar 18 2025

A223836 Number of n X 6 0..1 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

Original entry on oeis.org

22, 148, 503, 1286, 2884, 5992, 11749, 21912, 39064, 66854, 110269, 175938, 272468, 410812, 604669, 870916, 1230072, 1706794, 2330405, 3135454, 4162308, 5457776, 7075765, 9077968, 11534584, 14525070, 18138925, 22476506, 27649876, 33783684

Offset: 1

R. H. Hardin, Mar 27 2013

nonn

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

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

Column 6 of A223838.

Empirical: a(n) = (2/45)*n^6 + (55/36)*n^4 + (14/3)*n^3 + (3767/180)*n^2 + (293/6)*n - 116 for n>3.
Conjectures from Colin Barker, Aug 23 2018: (Start)
G.f.: x*(22 - 6*x - 71*x^2 + 103*x^3 + 35*x^4 - 77*x^5 + 10*x^6 + 22*x^7 - 4*x^8 - 2*x^9) / (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>10.
(End)

Name corrected by Andrew Howroyd, Mar 18 2025

A223837 Number of n X 7 0..1 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

Original entry on oeis.org

29, 239, 926, 2578, 6159, 13582, 28369, 56607, 108282, 199047, 352486, 602938, 998945, 1607388, 2518375, 3850945, 5759652, 8442093, 12147444, 17186068, 23940259, 32876186, 44557101, 59657875, 78980926, 103473603, 134247090, 172596894, 220024981, 278263624, 349301027

Offset: 1

R. H. Hardin, Mar 27 2013

nonn

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

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

Column 7 of A223838.

Empirical: a(n) = (4/315)*n^7 - (1/45)*n^6 + (28/45)*n^5 + (113/72)*n^4 + (2137/180)*n^3 + (14023/360)*n^2 + (13159/140)*n - 339 for n>4.
Conjectures from Colin Barker, Aug 23 2018: (Start)
G.f.: x*(29 + 7*x - 174*x^2 + 238*x^3 + 109*x^4 - 256*x^5 + 45*x^6 + 111*x^7 - 27*x^8 - 26*x^9 + 6*x^10 + 2*x^11) / (1 - x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n>12.
(End)

Name corrected by Andrew Howroyd, Mar 18 2025

A223839 Number of 3 X n 0..1 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

Original entry on oeis.org

4, 16, 48, 118, 255, 503, 926, 1614, 2690, 4318, 6712, 10146, 14965, 21597, 30566, 42506, 58176, 78476, 104464, 137374, 178635, 229891, 293022, 370166, 463742, 576474, 711416, 871978, 1061953, 1285545, 1547398, 1852626, 2206844, 2616200, 3087408, 3627782, 4245271

Offset: 1

R. H. Hardin, Mar 27 2013

nonn

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

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

Row 3 of A223838.

Empirical: a(n) = (1/720)*n^6 + (1/240)*n^5 + (29/144)*n^4 + (11/48)*n^3 + (1007/360)*n^2 - (37/30)*n + 2.
Conjectures from Colin Barker, Aug 23 2018: (Start)
G.f.: x*(2 - 2*x + x^2)*(2 - 4*x + 5*x^2 - 4*x^3 + 2*x^4) / (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>7.
(End)

Name corrected by Andrew Howroyd, Mar 18 2025

A223840 Number of 4 X n 0..1 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

Original entry on oeis.org

5, 25, 89, 249, 596, 1286, 2578, 4886, 8851, 15439, 26072, 42800, 68523, 107273, 164567, 247843, 366992, 535000, 768715, 1089755, 1525574, 2110704, 2888192, 3911252, 5245153, 6969365, 9179986, 11992474, 15544709, 20000411, 25552941, 32429513, 40895846, 51261286

Offset: 1

R. H. Hardin, Mar 27 2013

nonn

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

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

Row 4 of A223838.

Empirical: a(n) = (1/40320)*n^8 - (1/10080)*n^7 + (19/2880)*n^6 + (7/180)*n^5 + (527/5760)*n^4 + (3683/1440)*n^3 + (4051/10080)*n^2 - (1707/280)*n + 13 for n>2.
Conjectures from Colin Barker, Aug 24 2018: (Start)
G.f.: x*(5 - 20*x + 44*x^2 - 72*x^3 + 89*x^4 - 70*x^5 + 28*x^6 - 4*x^7 + 4*x^8 - 4*x^9 + x^10) / (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>11.
(End)

Name corrected by Andrew Howroyd, Mar 20 2025

A223841 Number of 5 X n 0..1 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

Original entry on oeis.org

6, 36, 149, 471, 1240, 2884, 6159, 12371, 23716, 43790, 78342, 136368, 231677, 385101, 627571, 1004341, 1580713, 2449699, 3742152, 5640008, 8393406, 12342594, 17945687, 25813519, 36753026, 51820812, 72388786, 100224016, 137585227, 187338675, 253096459, 339380689

Offset: 1

R. H. Hardin, Mar 27 2013

nonn

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

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

Row 5 of A223838.

Empirical: a(n) = (1/3628800)*n^10 - (1/241920)*n^9 + (17/120960)*n^8 - (1/13440)*n^7 + (3733/172800)*n^6 - (893/11520)*n^5 + (178931/90720)*n^4 - (188123/60480)*n^3 + (504149/25200)*n^2 - (30581/420)*n + 121 for n>4.

Name corrected by Andrew Howroyd, Mar 18 2025

cp's OEIS Frontend

A224173 T(n,k) = number of n X k 0..3 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

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A224262 T(n,k) = number of n X k 0..2 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

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A223833 Number of n X 3 0..1 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

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A223834 Number of n X 4 0..1 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

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A223835 Number of n X 5 0..1 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

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A223836 Number of n X 6 0..1 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

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A223837 Number of n X 7 0..1 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

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A223839 Number of 3 X n 0..1 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

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A223840 Number of 4 X n 0..1 arrays with rows, diagonals and antidiagonals unimodal and columns nondecreasing.

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