A252930 -id:A252930 - 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.

A252976 T(n,k) = Number of n X k nonnegative integer arrays with upper left 0 and lower right n+k-5 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 4, 13, 18, 13, 4, 10, 61, 153, 153, 61, 10, 20, 192, 770, 1236, 770, 192, 20, 35, 483, 2859, 6997, 6997, 2859, 483, 35, 56, 1050, 8694, 30802, 46812, 30802, 8694, 1050, 56, 84, 2058, 22924, 112877, 248182, 248182, 112877, 22924, 2058

Offset: 1

R. H. Hardin, Dec 25 2014

Table starts
..0....0......0.......1........4........10.........20..........35...........56
..0....0......1......13.......61.......192........483........1050.........2058
..0....1.....18.....153......770......2859.......8694.......22924........54272
..1...13....153....1236.....6997.....30802.....112877......359550......1024773
..4...61....770....6997....46812....248182....1100210.....4230324.....14477724
.10..192...2859...30802...248182...1592348....8528422....39423196....161160206
.20..483...8694..112877..1100210...8528422...54926890...303382053...1471499970
.35.1050..22924..359550..4230324..39423196..303382053..1988261908..11360377192
.56.2058..54272.1024773.14477724.161160206.1471499970.11360377192..75922639116
.84.3732.118057.2667554.44951694.593478797.6383377435.57644900961.447545856560

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

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), subtable T_{n X m}(n+m-5).

Cf. A252876, A252930. Column 1 is A000292(n-3). Cf. A252970-A252975 (columns 2-7).

Empirical for column k:
k=1: a(n) = (1/6)*n^3 - 1*n^2 + (11/6)*n - 1
k=2: [polynomial of degree 6]
k=3: [polynomial of degree 9]
k=4: [polynomial of degree 12]
k=5: [polynomial of degree 15]
k=6: [polynomial of degree 18]
k=7: [polynomial of degree 21]
Empirical: with "n+k-3" instead of "n+k-5" T(n,k) = binomial(n+k,k) - 2, see A166810, A166812, A166813.

A252923 Number of n X n nonnegative integer arrays with upper left 0 and lower right n+n-6 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

0, 0, 1, 615, 84910, 8241540, 759337545, 73117894428, 7578889491370, 848729993032718, 102072779621746876, 13069744040275286234, 1766655687081261435255, 250232365919081938956865, 36910187948533534644694094

Offset: 1

R. H. Hardin, Dec 24 2014

nonn

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

Alois P. Heinz, Table of n, a(n) for n = 1..20 (first 15 terms from R. H. Hardin)

Diagonal of A252930.

A252924 Number of n X 2 nonnegative integer arrays with upper left 0 and lower right n+2-6 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

0, 0, 0, 1, 19, 120, 483, 1500, 3923, 9069, 19095, 37356, 68860, 120835, 203424, 330525, 520794, 798830, 1196562, 1754859, 2525385, 3572722, 4976785, 6835554, 9268149, 12418275, 16458065, 21592350, 28063386, 36156069, 46203670, 58594123

Offset: 1

R. H. Hardin, Dec 24 2014

nonn

Column 2 of A252930.

			Some solutions for n=6:
..0..0....0..0....0..1....0..0....0..0....0..1....0..0....0..0....0..1....0..0
..0..1....1..1....1..1....1..1....0..0....0..1....0..1....0..0....0..1....1..1
..1..1....1..1....1..1....2..2....1..1....0..1....0..1....0..1....0..1....1..1
..1..1....1..1....1..2....2..2....1..1....0..1....1..1....1..1....0..1....1..2
..1..2....1..1....2..2....2..2....1..2....0..1....1..1....1..1....1..1....2..2
..2..2....1..2....2..2....2..2....1..2....1..2....2..2....1..2....1..2....2..2

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

Empirical: a(n) = (1/40320)*n^8 + (1/1120)*n^7 + (7/2880)*n^6 - (1/16)*n^5 + (647/5760)*n^4 + (87/160)*n^3 - (21317/10080)*n^2 + (141/56)*n - 1.

Empirical: G.f.: x^4*(-1-10*x+15*x^2-3*x^3-3*x^4+x^5) / (x-1)^9 . - R. J. Mathar, Nov 21 2015

A252929 Number of nX7 nonnegative integer arrays with upper left 0 and lower right n+7-6 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

15, 483, 13976, 315198, 5385305, 71297441, 759337545, 6725497344, 50869309436, 335549742230, 1963313662947, 10332594393296, 49485574097413, 217813226273441, 888570804281664, 3384240524056852, 12109669540486657

Offset: 1

R. H. Hardin, Dec 24 2014

nonn

Column 7 of A252930

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

R. H. Hardin, Table of n, a(n) for n = 1..210
R. H. Hardin, Empirical polynomial of degree 28

Empirical polynomial of degree 28 (see link above)

A252925 Number of n X 3 nonnegative integer arrays with upper left 0 and lower right n+3-6 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

0, 0, 1, 33, 413, 2859, 13976, 54199, 177848, 513905, 1342933, 3233784, 7275049, 15451358, 31234528, 60485192, 112793064, 203424686, 356097793, 606862907, 1009447181, 1642504781, 2619324320, 4100669333, 6311574992, 9563095895

Offset: 1

R. H. Hardin, Dec 24 2014

nonn

Column 3 of A252930.

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

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

Empirical: a(n) = (1/59875200)*n^12 + (1/997920)*n^11 + (31/1360800)*n^10 + (1/5040)*n^9 + (79/453600)*n^8 - (71/15120)*n^7 - (4127/1360800)*n^6 + (319/9072)*n^5 + (52061/5443200)*n^4 - (2587/6048)*n^3 + (3315209/1663200)*n^2 - (11097/3080)*n + 2.

Empirical: G.f.: -x^3*(1 +20*x +62*x^2 -222*x^3 +300*x^4 -297*x^5 +255*x^6 -167*x^7 +72*x^8 -18*x^9 +2*x^10) / (x-1)^13. - R. J. Mathar, Nov 21 2015

A252926 Number of n X 4 nonnegative integer arrays with upper left 0 and lower right n+4-6 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

0, 1, 33, 615, 6997, 53950, 315198, 1499394, 6083808, 21733215, 69921908, 206052448, 563440284, 1444285973, 3499040581, 8065603025, 17787652381, 37705694442, 77126966988, 152747362302, 293741640632, 549886035657

Offset: 1

R. H. Hardin, Dec 24 2014

nonn

Column 4 of A252930.

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

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

Cf. A252930.

Empirical: a(n) = (13/1609445376000)*n^16 + (13/18289152000)*n^15 + (3947/149448499200)*n^14 + (20057/37362124800)*n^13 + (728009/114960384000)*n^12 + (312437/7185024000)*n^11 + (1303193/7315660800)*n^10 + (848039/1828915200)*n^9 + (11500417/20901888000)*n^8 - (1346209/373248000)*n^7 - (14478857/1149603840)*n^6 - (361673/8164800)*n^5 + (13716552599/33530112000)*n^4 + (454863863/1397088000)*n^3 - (207327073/86486400)*n^2 + (137883/80080)*n.

Empirical: G.f.: x^2*(-1 -16*x -190*x^2 -350*x^3 +1419*x^4 -3792*x^5 +7860*x^6 -11238*x^7 +11247*x^8 -8133*x^9 +4280*x^10 -1600*x^11 +401*x^12 -60*x^13+4*x^14) / (x-1)^17. - R. J. Mathar, Nov 23 2015

A252927 Number of nX5 nonnegative integer arrays with upper left 0 and lower right n+5-6 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

1, 19, 413, 6997, 84910, 762227, 5385305, 31454256, 157376166, 692347393, 2731553014, 9814551889, 32510650689, 100275435009, 290361386801, 794754882094, 2068183070502, 5142171510385, 12267196687798, 28182388823719

Offset: 1

R. H. Hardin, Dec 24 2014

nonn

Column 5 of A252930

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

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

Empirical: a(n) = (47/17888985354240000)*n^20 + (47/149074877952000)*n^19 + (167/10003708915200)*n^18 + (4591/8892185702400)*n^17 + (29063/2802159360000)*n^16 + (187361/1307674368000)*n^15 + (13481939/9415255449600)*n^14 + (1082533/98075577600)*n^13 + (454942751/6584094720000)*n^12 + (23035147/67060224000)*n^11 + (43650403/34488115200)*n^10 + (134989213/40236134400)*n^9 + (2353290548929/470762772480000)*n^8 - (1781952749/3923023104000)*n^7 + (174219179149/2353813862400)*n^6 + (1662462491/16345929600)*n^5 + (42818968289/66718080000)*n^4 - (16827599/22422400)*n^3 + (9481442611/5333065920)*n^2 - (565043267/116396280)*n + 4.

Empirical: G.f.: -x*(1 -2*x +224*x^2 +984*x^3 +5418*x^4 -7437*x^5 +43066*x^6 -112135*x^7 +198529*x^8 -285030*x^9 +339839*x^10 -332484*x^11 +266120*x^12 -174781*x^13 +93817*x^14 -40498*x^15 +13679*x^16 -3477*x^17 +627*x^18 -72*x^19 +4*x^20) / (x-1)^21 . - R. J. Mathar, Nov 24 2015

A252928 Number of nX6 nonnegative integer arrays with upper left 0 and lower right n+6-6 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

5, 120, 2859, 53950, 762227, 8241540, 71297441, 512868867, 3160111147, 17063990547, 82211677028, 358529577294, 1432148475494, 5291678799968, 18236349617760, 59031011794076, 180569195179185, 524679580969208

Offset: 1

R. H. Hardin, Dec 24 2014

nonn

Column 6 of A252930

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

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

Empirical: a(n) = (331/583128197117706240000)*n^24 + (331/3738001263575040000)*n^23 + (31687/5070679974936576000)*n^22 + (9719/36584992604160000)*n^21 + (83457079/10948059036794880000)*n^20 + (2293999511/14597412049059840000)*n^19 + (70133801/28810681675776000)*n^18 + (11352513167/384142422343680000)*n^17 + (39752887/135444234240000)*n^16 + (18525490537/7532204359680000)*n^15 + (1041895258531/59655058528665600)*n^14 + (51595236448757/497125487738880000)*n^13 + (383722831905761/745688231608320000)*n^12 + (7387354289669/3476402012160000)*n^11 + (96126694933271/13557967847424000)*n^10 + (3074811780539/156920924160000)*n^9 + (7254553057678669/144053408378880000)*n^8 + (5122046760654749/48017802792960000)*n^7 + (615467828621/1796401152000)*n^6 + (481180027609747/666913927680000)*n^5 + (6796574742313/5225109120000)*n^4 + (10424925937303/10188962784000)*n^3 - (3393867242681/1060137318240)*n^2 + (7032359801/2677114440)*n + 2

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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A252976 T(n,k) = Number of n X k nonnegative integer arrays with upper left 0 and lower right n+k-5 and value increasing by 0 or 1 with every step right or down.

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A252923 Number of n X n nonnegative integer arrays with upper left 0 and lower right n+n-6 and value increasing by 0 or 1 with every step right or down.

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A252924 Number of n X 2 nonnegative integer arrays with upper left 0 and lower right n+2-6 and value increasing by 0 or 1 with every step right or down.

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A252929 Number of nX7 nonnegative integer arrays with upper left 0 and lower right n+7-6 and value increasing by 0 or 1 with every step right or down.

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A252925 Number of n X 3 nonnegative integer arrays with upper left 0 and lower right n+3-6 and value increasing by 0 or 1 with every step right or down.

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A252926 Number of n X 4 nonnegative integer arrays with upper left 0 and lower right n+4-6 and value increasing by 0 or 1 with every step right or down.

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A252927 Number of nX5 nonnegative integer arrays with upper left 0 and lower right n+5-6 and value increasing by 0 or 1 with every step right or down.

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A252928 Number of nX6 nonnegative integer arrays with upper left 0 and lower right n+6-6 and value increasing by 0 or 1 with every step right or down.

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