A252876 -id:A252876 - cp's OEIS frontend

A323846 Array read by antidiagonals: T(m,n) = number of m X n matrices M with entries {0,1,2} that have M_{1,1}=0, M_{m,n}=2, are such that the rows and columns are monotonic without jumps of 2, and satisfy M_{(i+1),(j+1)} = M_{i,j} + (0 or 1).

0, 0, 0, 1, 0, 1, 3, 4, 4, 3, 6, 16, 25, 16, 6, 10, 41, 94, 94, 41, 10, 15, 85, 266, 386, 266, 85, 15, 21, 155, 632, 1247, 1247, 632, 155, 21, 28, 259, 1332, 3423, 4657, 3423, 1332, 259, 28, 36, 406, 2570, 8342, 14795, 14795, 8342, 2570, 406, 36, 45, 606, 4631, 18546, 41586, 54219, 41586, 18546, 4631, 606, 45

Offset: 1

N. J. A. Sloane, Feb 06 2019

The monotonicity condition requires that M_{(i+1),j} = M_{i,j} + (0 or 1); M_{i,(j+1)} = M_{i,j} + (0 or 1).
These matrices can be cut into three connected pieces, containing the 0's, 1's, and 2's; there are two vertex-disjoint paths from the north-and-east edges of the matrix to the south-and-west edges.
Row (or column) n >= 1 has a linear recurrence (with constant coefficients) of order 2n+1. - Alois P. Heinz, Feb 07 2019

			Array begins:
    0   0    1    3     6    10 ...
    0   0    4   16    41    85 ...
    1   4   25   94   266   632 ...
    3  16   94  386  1247  3423 ...
    6  41  266 1247  4657 14795 ...
   10  85  632 3427 14795 54219 ...
...
The 4 examples when m=2 and n=3 are
    011   011  012   012
    012   112  012   112

D. E. Knuth, Email to N. J. A. Sloane, Feb 05 2019.

Alois P. Heinz, Antidiagonals n = 1..80, flattened

Rows 1-10 give: A000217(n-2), A323847, A323967, A323968, A323969, A323970, A323971, A323972, A323973, A323974.
Main diagonal gives A306322.
Cf. A132823, A252876, A229428.

More terms from Alois P. Heinz, Feb 07 2019

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.

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.

A252869 Number of n X n nonnegative integer arrays with upper left 0 and lower right 2n-4 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

0, 1, 44, 615, 7313, 85801, 1030330, 12742873, 161937617, 2105918045, 27918122131, 376111502977, 5136280530303, 70961632168027, 990271171218092, 13940676224604419, 197768060575469207, 2824828302263709187, 40595088350375470407, 586585037283553891311

Offset: 1

R. H. Hardin, Dec 24 2014

nonn

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

Main diagonal of A252876.

Column d=2 of A323849.

a(16)-a(20) from Alois P. Heinz, Feb 07 2019

A252870 Number of n X 2 nonnegative integer arrays with upper left 0 and lower right n+2-4 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

0, 1, 8, 26, 61, 120, 211, 343, 526, 771, 1090, 1496, 2003, 2626, 3381, 4285, 5356, 6613, 8076, 9766, 11705, 13916, 16423, 19251, 22426, 25975, 29926, 34308, 39151, 44486, 50345, 56761, 63768, 71401, 79696, 88690, 98421, 108928, 120251, 132431, 145510

Offset: 1

R. H. Hardin, Dec 24 2014

nonn

Column 2 of A252876.

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

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

Empirical: a(n) = (1/24)*n^4 + (5/12)*n^3 - (13/24)*n^2 - (11/12)*n + 1.

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

A252871 Number of nX3 nonnegative integer arrays with upper left 0 and lower right n+3-4 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

1, 8, 44, 153, 413, 949, 1948, 3676, 6497, 10894, 17492, 27083, 40653, 59411, 84820, 118630, 162913, 220100, 293020, 384941, 499613, 641313, 814892, 1025824, 1280257, 1585066, 1947908, 2377279, 2882573, 3474143, 4163364, 4962698, 5885761, 6947392

Offset: 1

R. H. Hardin, Dec 24 2014

nonn

Column 3 of A252876

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

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

Empirical: a(n) = (1/360)*n^6 + (1/20)*n^5 + (5/18)*n^4 + (1/3)*n^3 + (79/360)*n^2 - (113/60)*n + 2.

Empirical: G.f.: -x*(1+x+9*x^2-22*x^3+21*x^4-10*x^5+2*x^6) / (x-1)^7. - R. J. Mathar, Nov 23 2015

A252872 Number of n X 4 nonnegative integer arrays with upper left 0 and lower right n+4-4 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

3, 26, 153, 615, 1953, 5281, 12686, 27805, 56624, 108549, 197804, 345216, 580452, 944778, 1494415, 2304572, 3474241, 5131844, 7441827, 10612301, 14903835, 20639511, 28216356, 38118271, 50930582, 67356343, 88234526, 114560238

Offset: 1

R. H. Hardin, Dec 24 2014

nonn

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

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

Column 4 of A252876.

Empirical: a(n) = (1/8064)*n^8 + (1/288)*n^7 + (103/2880)*n^6 + (17/90)*n^5 + (655/1152)*n^4 + (331/288)*n^3 - (677/1120)*n^2 - (41/120)*n + 2.

Empirical: G.f.: -x*(3-x+27*x^2-78*x^3+120*x^4-110*x^5+59*x^6-17*x^7+2*x^8) / (x-1)^9. - R. J. Mathar, Nov 23 2015

A252873 Number of n X 5 nonnegative integer arrays with upper left 0 and lower right n+5-4 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

6, 61, 413, 1953, 7313, 23203, 64920, 164399, 383735, 836797, 1722394, 3373453, 6327844, 11427851, 19956855, 33823579, 55806260, 89871373, 141584051, 218630137, 331472883, 494170691, 725385986, 1049619335, 1498707293

Offset: 1

R. H. Hardin, Dec 24 2014

nonn

Column 5 of A252876.

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

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

Empirical: a(n) = (1/259200)*n^10 + (1/6480)*n^9 + (149/60480)*n^8 + (163/7560)*n^7 + (10411/86400)*n^6 + (1027/2160)*n^5 + (16357/12960)*n^4 + (937/810)*n^3 + (1756/1575)*n^2 - (727/630)*n + 3.

Empirical: G.f.: -x*(6 -5*x +72*x^2 -225*x^3 +460*x^4 -612*x^5 +537*x^6 -315*x^7 +121*x^8 -28*x^9 +3*x^10) / (x-1)^11. - R. J. Mathar, Nov 21 2015

A252874 Number of n X 6 nonnegative integer arrays with upper left 0 and lower right n+6-4 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

10, 120, 949, 5281, 23203, 85801, 277585, 806347, 2142634, 5281314, 12207123, 26687363, 55571337, 110851465, 212839721, 394940091, 710658874, 1243696244, 2122211729, 3538661109, 5776971495, 9249264713, 14544867193, 22494968931

Offset: 1

R. H. Hardin, Dec 24 2014

nonn

Column 6 of A252876.

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

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

Empirical: a(n) = (1/11404800)*n^12 + (1/211200)*n^11 + (37/345600)*n^10 + (143/103680)*n^9 + (27977/2419200)*n^8 + (83399/1209600)*n^7 + (318253/1036800)*n^6 + (6655/6912)*n^5 + (17647/9600)*n^4 + (350243/129600)*n^3 + (95119/277200)*n^2 + (5297/6930)*n + 3.

Empirical: G.f.: -x*(10 -10*x +169*x^2 -556*x^3 +1402*x^4 -2404*x^5 +2895*x^6 -2526*x^7 +1599*x^8 -716*x^9 +214*x^10 -38*x^11 +3*x^12) / (x-1)^13. - R. J. Mathar, Nov 21 2015

A252875 Number of nX7 nonnegative integer arrays with upper left 0 and lower right n+7-4 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

15, 211, 1948, 12686, 64920, 277585, 1030330, 3407823, 10237249, 28340232, 73128376, 177500233, 408299906, 895580310, 1882850350, 3810729118, 7452433875, 14127861337, 26034391284, 46747817584, 81967052716, 140602733711

Offset: 1

R. H. Hardin, Dec 24 2014

nonn

Column 7 of A252876

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

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

Empirical: a(n) = (1/660441600)*n^14 + (1/9434880)*n^13 + (1/311850)*n^12 + (283/4989600)*n^11 + (23/34560)*n^10 + (5/896)*n^9 + (55843/1587600)*n^8 + (153793/907200)*n^7 + (4422487/7257600)*n^6 + (1152217/725760)*n^5 + (14953/4800)*n^4 + (834067/302400)*n^3 + (46071121/16816800)*n^2 - (359/17160)*n + 4

cp's OEIS Frontend

A323846 Array read by antidiagonals: T(m,n) = number of m X n matrices M with entries {0,1,2} that have M_{1,1}=0, M_{m,n}=2, are such that the rows and columns are monotonic without jumps of 2, and satisfy M_{(i+1),(j+1)} = M_{i,j} + (0 or 1).

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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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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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A252869 Number of n X n nonnegative integer arrays with upper left 0 and lower right 2n-4 and value increasing by 0 or 1 with every step right or down.

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A252870 Number of n X 2 nonnegative integer arrays with upper left 0 and lower right n+2-4 and value increasing by 0 or 1 with every step right or down.

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A252871 Number of nX3 nonnegative integer arrays with upper left 0 and lower right n+3-4 and value increasing by 0 or 1 with every step right or down.

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A252872 Number of n X 4 nonnegative integer arrays with upper left 0 and lower right n+4-4 and value increasing by 0 or 1 with every step right or down.

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A252873 Number of n X 5 nonnegative integer arrays with upper left 0 and lower right n+5-4 and value increasing by 0 or 1 with every step right or down.

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A252874 Number of n X 6 nonnegative integer arrays with upper left 0 and lower right n+6-4 and value increasing by 0 or 1 with every step right or down.

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A252875 Number of nX7 nonnegative integer arrays with upper left 0 and lower right n+7-4 and value increasing by 0 or 1 with every step right or down.

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