A252976 -id:A252976 - cp's OEIS frontend

A252876 T(n,k) = Number of n X k nonnegative integer arrays with upper left 0 and lower right n+k-4 and value increasing by 0 or 1 with every step right or down.

0, 0, 0, 1, 1, 1, 3, 8, 8, 3, 6, 26, 44, 26, 6, 10, 61, 153, 153, 61, 10, 15, 120, 413, 615, 413, 120, 15, 21, 211, 949, 1953, 1953, 949, 211, 21, 28, 343, 1948, 5281, 7313, 5281, 1948, 343, 28, 36, 526, 3676, 12686, 23203, 23203, 12686, 3676, 526, 36, 45, 771, 6497, 27805, 64920, 85801, 64920, 27805, 6497, 771, 45

Offset: 1

R. H. Hardin, Dec 24 2014

Table starts
..0...0.....1......3......6......10.......15........21........28.........36
..0...1.....8.....26.....61.....120......211.......343.......526........771
..1...8....44....153....413.....949.....1948......3676......6497......10894
..3..26...153....615...1953....5281....12686.....27805.....56624.....108549
..6..61...413...1953...7313...23203....64920....164399....383735.....836797
.10.120...949...5281..23203...85801...277585....806347...2142634....5281314
.15.211..1948..12686..64920..277585..1030330...3407823..10237249...28340232
.21.343..3676..27805.164399..806347..3407823..12742873..42993671..132872804
.28.526..6497..56624.383735.2142634.10237249..42993671.161937617..555632319
.36.771.10894.108549.836797.5281314.28340232.132872804.555632319.2105918045

			Some solutions for n=3 k=4
..0..1..1..1....0..0..1..1....0..1..2..3....0..0..1..1....0..0..1..1
..1..1..2..2....0..1..1..2....1..1..2..3....0..0..1..2....0..1..2..2
..1..1..2..3....1..2..2..3....1..2..2..3....1..1..2..3....1..1..2..3

Alois P. Heinz, Table of n, a(n) for n = 1..2850 (first 479 terms from R. H. Hardin)
R. J. Mathar, Counting 2-way monotonic terrace forms over rectangular landscapes, vixra 1511.0225 (2015)

Columns 1-7 give: A000217(n-2), A252870, A252871, A252872, A252873, A252874, A252875.
Main diagonal is A252869.
Cf. A252976, A252930, A323846.

Empirical for column k:
k=1: a(n) = (1/2)*n^2 - (3/2)*n + 1
k=2: a(n) = (1/24)*n^4 + (5/12)*n^3 - (13/24)*n^2 - (11/12)*n + 1,
k=3: [polynomial of degree 6]
k=4: [polynomial of degree 8]
k=5: [polynomial of degree 10]
k=6: [polynomial of degree 12]
k=7: [polynomial of degree 14]
Empirical: with "n+k-3" instead of "n+k-4" T(n,k) = binomial(n+k,k) - 2.

A252930 T(n,k) = Number of n X k nonnegative integer arrays with upper left 0 and lower right n+k-6 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 5, 19, 33, 33, 19, 5, 15, 120, 413, 615, 413, 120, 15, 35, 483, 2859, 6997, 6997, 2859, 483, 35, 70, 1500, 13976, 53950, 84910, 53950, 13976, 1500, 70, 126, 3923, 54199, 315198, 762227, 762227, 315198, 54199, 3923, 126, 210

Offset: 1

R. H. Hardin, Dec 24 2014

Table starts
...0....0......0........0.........1...........5...........15............35
...0....0......0........1........19.........120..........483..........1500
...0....0......1.......33.......413........2859........13976.........54199
...0....1.....33......615......6997.......53950.......315198.......1499394
...1...19....413.....6997.....84910......762227......5385305......31454256
...5..120...2859....53950....762227.....8241540.....71297441.....512868867
..15..483..13976...315198...5385305....71297441....759337545....6725497344
..35.1500..54199..1499394..31454256...512868867...6725497344...73117894428
..70.3923.177848..6083808.157376166..3160111147..50869309436..675539536773
.126.9069.513905.21733215.692347393.17063990547.335549742230.5411459549576

			Some solutions for n=4, k=4:
..0..1..1..1....0..0..0..0....0..0..0..0....0..1..1..2....0..0..0..1
..1..1..1..1....1..1..1..1....0..0..0..1....1..1..2..2....0..1..1..2
..1..1..2..2....1..1..1..1....1..1..1..1....1..1..2..2....0..1..1..2
..1..2..2..2....1..1..1..2....1..2..2..2....1..2..2..2....1..2..2..2

R. H. Hardin, Table of n, a(n) for n = 1..449
R. J. Mathar, Counting 2-way monotonic terrace forms over rectangular landscapes, vixra 1511.0225 (2015)

Column 1 is A000332(n-1), other columns are A252924 - A252929. Cf. A252923 (diagonal); A252876 (lower right n+k-4), A252976 (lower right n+k-5).

Empirical for column k:
k=1: a(n) = (1/24)*n^4 - (5/12)*n^3 + (35/24)*n^2 - (25/12)*n + 1.
k=2: [polynomial of degree 8]
k=3: [polynomial of degree 12]
k=4: [polynomial of degree 16]
k=5: [polynomial of degree 20]
k=6: [polynomial of degree 24]
k=7: [polynomial of degree 28]
Empirical: with "n+k-3" instead of "n+k-6" T(n,k) = binomial(n+k,k) - 2.

A252970 Number of nX2 nonnegative integer arrays with upper left 0 and lower right n+2-5 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

0, 0, 1, 13, 61, 192, 483, 1050, 2058, 3732, 6369, 10351, 16159, 24388, 35763, 51156, 71604, 98328, 132753, 176529, 231553, 299992, 384307, 487278, 612030, 762060, 941265, 1153971, 1404963, 1699516, 2043427, 2443048, 2905320, 3437808, 4048737

Offset: 1

R. H. Hardin, Dec 25 2014

nonn

Column 2 of A252976

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

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

Empirical: a(n) = (1/720)*n^6 + (7/240)*n^5 - (1/144)*n^4 - (31/48)*n^3 + (271/180)*n^2 - (53/60)*n = n*(n-1)*(n-2)*(n^3+24*n^2+65*n-318)/720.

Empirical: G.f.: -x^3*(1+6*x-9*x^2+3*x^3) / (x-1)^7 . - R. J. Mathar, Nov 23 2015

A252975 Number of nX7 nonnegative integer arrays with upper left 0 and lower right n+7-5 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

20, 483, 8694, 112877, 1100210, 8528422, 54926890, 303382053, 1471499970, 6383377435, 25130419118, 90859574359, 304675148476, 955390407363, 2821209579406, 7892066379909, 21022079637900, 53557533613173

Offset: 1

R. H. Hardin, Dec 25 2014

nonn

Column 7 of A252976

			Some solutions for n=4
..0..1..2..3..3..3..4....0..1..1..2..2..3..4....0..0..1..2..3..3..3
..1..2..2..3..3..3..4....1..1..2..3..3..4..5....1..1..1..2..3..4..4
..2..2..2..3..3..4..5....2..2..3..3..4..5..6....1..1..2..2..3..4..5
..2..3..3..4..4..5..6....2..3..4..4..5..6..6....1..1..2..3..4..5..6

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

Empirical: a(n) = (491/510909421717094400)*n^21 + (491/4054836680294400)*n^20 + (31237/4561691265331200)*n^19 + (16931/72754246656000)*n^18 + (6860417/1280474741145600)*n^17 + (5620079/62768369664000)*n^16 + (28238557/24715045555200)*n^15 + (460147/39626496000)*n^14 + (1994012479/20542375526400)*n^13 + (9803403773/14485008384000)*n^12 + (2842330237/724250419200)*n^11 + (22704330871/1207084032000)*n^10 + (117780405261823/1581762915532800)*n^9 + (1372309011587/5706215424000)*n^8 + (17399771450683/28245766348800)*n^7 + (29498769372031/23538138624000)*n^6 + (57797311549729/26676557107200)*n^5 + (6071375333369/2020951296000)*n^4 + (7008309815167/1119943843200)*n^3 + (375662307017/83805321600)*n^2 + (152121493/25865840)*n - 4

A252971 Number of nX3 nonnegative integer arrays with upper left 0 and lower right n+3-5 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

0, 1, 18, 153, 770, 2859, 8694, 22924, 54272, 118057, 239798, 460207, 841930, 1478451, 2505634, 4116442, 6579440, 10261761, 15657290, 23420901, 34409666, 49732043, 70806142, 99428264, 137853008, 188886345, 255993166, 343420923

Offset: 1

R. H. Hardin, Dec 25 2014

nonn

Column 3 of A252976.

			Some solutions for n=4
..0..1..1....0..0..1....0..1..1....0..1..1....0..0..1....0..0..0....0..0..1
..0..1..2....0..0..1....1..1..2....0..1..2....0..1..2....0..1..1....0..0..1
..1..1..2....0..1..2....1..1..2....1..2..2....1..1..2....1..1..1....1..1..1
..2..2..2....1..2..2....1..1..2....1..2..2....1..2..2....1..1..2....2..2..2

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

Empirical: a(n) = (1/90720)*n^9 + (1/2520)*n^8 + (19/3780)*n^7 + (7/360)*n^6 - (59/4320)*n^5 - (19/180)*n^4 + (5827/45360)*n^3 - (32/35)*n^2 + (3629/1260)*n - 2.

Empirical: G.f.: x^2*(1+8*x+18*x^2-70*x^3+94*x^4-78*x^5+43*x^6-14*x^7+2*x^8) / (x-1)^10. - R. J. Mathar, Nov 23 2015

A252972 Number of nX4 nonnegative integer arrays with upper left 0 and lower right n+4-5 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

1, 13, 153, 1236, 6997, 30802, 112877, 359550, 1024773, 2667554, 6438457, 14575914, 31230978, 63789948, 124931040, 235737421, 430298389, 762367687, 1314817821, 2212837300, 3642069999, 5873199042, 9294839027, 14457028052, 22128113529

Offset: 1

R. H. Hardin, Dec 25 2014

nonn

Column 4 of A252976

			Some solutions for n=4
..0..1..1..1....0..1..2..3....0..0..0..1....0..1..2..2....0..0..1..2
..0..1..1..1....0..1..2..3....1..1..1..2....0..1..2..2....0..1..2..2
..0..1..2..2....1..2..2..3....2..2..2..2....1..1..2..3....1..2..2..2
..0..1..2..3....1..2..3..3....2..2..3..3....1..2..2..3....2..3..3..3

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

Empirical: a(n) = (29/479001600)*n^12 + (29/8870400)*n^11 + (3083/43545600)*n^10 + (233/290304)*n^9 + (73739/14515200)*n^8 + (48151/2419200)*n^7 + (2494049/43545600)*n^6 + (31657/483840)*n^5 - (1169489/10886400)*n^4 - (1024151/1814400)*n^3 + (2116423/831600)*n^2 - (601/27720)*n - 1

A252973 Number of nX5 nonnegative integer arrays with upper left 0 and lower right n+5-5 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

4, 61, 770, 6997, 46812, 248182, 1100210, 4230324, 14477724, 44951694, 128484004, 341970798, 855386126, 2025957453, 4571859908, 9881224135, 20544547580, 41246142411, 80218821614, 151563108019, 278867435440, 500751223536

Offset: 1

R. H. Hardin, Dec 25 2014

nonn

Column 5 of A252976

			Some solutions for n=4
..0..1..1..1..1....0..1..1..1..1....0..1..2..2..3....0..1..2..2..2
..0..1..2..2..2....0..1..2..2..2....1..1..2..2..3....0..1..2..2..3
..1..2..2..3..3....1..2..3..3..3....1..2..2..3..4....1..1..2..3..4
..1..2..2..3..4....2..3..3..3..4....2..2..3..4..4....2..2..3..4..4

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

Empirical: a(n) = (29/130767436800)*n^15 + (29/1743565824)*n^14 + (761/1437004800)*n^13 + (4567/479001600)*n^12 + (156803/1437004800)*n^11 + (12553/14515200)*n^10 + (4753991/914457600)*n^9 + (7470857/304819200)*n^8 + (5376871/65318400)*n^7 + (1929223/10886400)*n^6 + (43011707/179625600)*n^5 + (2111353/19958400)*n^4 + (648195787/454053600)*n^3 + (11630747/16816800)*n^2 + (39227/9240)*n - 3

A252974 Number of nX6 nonnegative integer arrays with upper left 0 and lower right n+6-5 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

10, 192, 2859, 30802, 248182, 1592348, 8528422, 39423196, 161160206, 593478797, 1997447386, 6215980425, 18056320047, 49345966862, 127715626611, 314800577783, 742509146839, 1682790560239, 3677643193138, 7774488966837, 15941255148133

Offset: 1

R. H. Hardin, Dec 25 2014

nonn

Column 6 of A252976

			Some solutions for n=4
..0..1..1..2..3..4....0..1..2..2..3..4....0..1..1..2..3..3....0..1..1..2..3..3
..0..1..2..3..4..4....1..2..3..3..3..4....0..1..2..2..3..4....1..1..2..2..3..3
..1..2..3..4..4..4....2..3..3..3..3..4....0..1..2..3..4..5....1..2..2..3..3..4
..2..3..3..4..4..5....3..3..3..4..4..5....1..2..3..4..4..5....2..2..2..3..4..5

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

Empirical: a(n) = (883/1600593426432000)*n^18 + (883/16167610368000)*n^17 + (148963/62768369664000)*n^16 + (6599/108972864000)*n^15 + (8020459/7846046208000)*n^14 + (3062099/249080832000)*n^13 + (813101243/7242504192000)*n^12 + (165163519/201180672000)*n^11 + (1074230093/219469824000)*n^10 + (564482173/24385536000)*n^9 + (410608176823/4828336128000)*n^8 + (212043001/870912000)*n^7 + (12296690400809/23538138624000)*n^6 + (525004330021/653837184000)*n^5 + (1880413300609/1307674368000)*n^4 + (10237127057/6054048000)*n^3 + (183696263083/30875644800)*n^2 + (15176011/12252240)*n - 2

A252969 Number of n X n nonnegative integer arrays with upper left 0 and lower right n+n-5 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

0, 0, 18, 1236, 46812, 1592348, 54926890, 1988261908, 75922639116, 3044977814280, 127417950149256, 5527784646050032, 247288999212565632, 11357451257590359840, 533618856431642996126, 25573783371050570978268, 1247199857452953231731592, 61772272799794134860411162

Offset: 1

R. H. Hardin, Dec 25 2014

nonn

Diagonal of A252976.

a(16)-a(18) from Alois P. Heinz, Feb 07 2019

cp's OEIS Frontend

A252876 T(n,k) = Number of n X k nonnegative integer arrays with upper left 0 and lower right n+k-4 and value increasing by 0 or 1 with every step right or down.

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A252930 T(n,k) = Number of n X k nonnegative integer arrays with upper left 0 and lower right n+k-6 and value increasing by 0 or 1 with every step right or down.

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A252970 Number of nX2 nonnegative integer arrays with upper left 0 and lower right n+2-5 and value increasing by 0 or 1 with every step right or down.

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A252975 Number of nX7 nonnegative integer arrays with upper left 0 and lower right n+7-5 and value increasing by 0 or 1 with every step right or down.

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A252971 Number of nX3 nonnegative integer arrays with upper left 0 and lower right n+3-5 and value increasing by 0 or 1 with every step right or down.

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A252972 Number of nX4 nonnegative integer arrays with upper left 0 and lower right n+4-5 and value increasing by 0 or 1 with every step right or down.

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A252973 Number of nX5 nonnegative integer arrays with upper left 0 and lower right n+5-5 and value increasing by 0 or 1 with every step right or down.

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A252974 Number of nX6 nonnegative integer arrays with upper left 0 and lower right n+6-5 and value increasing by 0 or 1 with every step right or down.

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A252969 Number of n X n nonnegative integer arrays with upper left 0 and lower right n+n-5 and value increasing by 0 or 1 with every step right or down.

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