A323846 -id:A323846 - 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.

A306322 Number of n X n integer matrices (m_{i,j}) such that m_{1,1}=0, m_{n,n}=2, and all rows, columns, and falling diagonals are (weakly) monotonic without jumps of 2.

Original entry on oeis.org

1, 0, 0, 25, 386, 4657, 54219, 642815, 7852836, 98755951, 1273299491, 16761968919, 224508932229, 3051075581019, 41979207169125, 583745779595077, 8192478969914858, 115908383594664493, 1651636256584103013, 23685002515500875105, 341589590792856093329

Offset: 0

Alois P. Heinz, Feb 07 2019

nonn

Manuel Kauers and Christoph Koutschan, Table of n, a(n) for n = 0..836 (terms 0..40 from Alois P. Heinz).
M. Kauers and C. Koutschan, Some D-finite and some possibly D-finite sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023.

Main diagonal of A323846.

Column d=2 of A323848.

Nara[i_, j_] := 1/(i+j-1)*Binomial[i+j-1, i]*Binomial[i+j-1, i-1];
Prepend[Table[2*Sum[Sum[Nara[i, j], {i, n}] + (n-j-1)*Nara[j, n], {j, n}] - 2*Binomial[2*n, n] + Nara[n, n] + 3, {n, 100}], 1] (* Manuel Kauers and Christoph Koutschan, Mar 02 2023 *)

From Manuel Kauers and Christoph Koutschan, Mar 02 2023: (Start)
a(n) = 2*Sum_{j=1..n} (Sum_{i=1..n} N'(i, j) + (n-j-1)*N'(j, n)) - 2*binomial(2*n, n) + N'(n, n) + 3 for n>0, where N'(n, k) = (binomial(n+k-1, n-1)*binomial(n+k-1, n))/(n+k-1) denotes the Narayana number N(n+k-1, k).
Recurrence: -2*(n+3)*(n+4)^2*(2*n+7)*(118125*n^10 + 1308375*n^9 + 6016950*n^8 + 14827410*n^7 + 20875365*n^6 + 15986367*n^5 + 4449768*n^4 - 2342808*n^3 - 2279152*n^2 - 660240*n - 64000)*a(n+4) + (n+3)*(10040625*n^13 + 210555000*n^12 + 1942194375*n^11 + 10361592450*n^10 + 35325144315*n^9 + 80085358620*n^8 + 121180651809*n^7 + 117919810482*n^6 + 64349576684*n^5 + 7017979960*n^4 - 14571577344*n^3 - 9566235392*n^2 - 2428639744*n - 221347840)*a(n+3) + (-42170625*n^14 - 981642375*n^13 - 10209053025*n^12 - 62559627795*n^11 - 250621464735*n^10 - 687475711989*n^9 - 1311094658043*n^8 - 1718884004625*n^7 - 1471227292164*n^6 - 691541238960*n^5 - 14462120192*n^4 + 188403075920*n^3 + 108128100864*n^2 + 25779317504*n + 2257059840)*a(n+2) + 2*(2*n+3)*(10040625*n^13 + 215870625*n^12 + 2048259375*n^11 + 11279217825*n^10 + 39828085965*n^9 + 93825035775*n^8 + 147951032109*n^7 + 150478534491*n^6 + 86482913102*n^5 + 11547320420*n^4 - 18788310824*n^3 - 12713618176*n^2 - 3178474112*n - 272670720)*a(n+1) - 8*n*(2*n-1)*(2*n+1)*(2*n+3)*(118125*n^10 + 2489625*n^9 + 23107950*n^8 + 124239510*n^7 + 427851585*n^6 + 984186117*n^5 + 1527319428*n^4 + 1572814284*n^3 + 1022652512*n^2 + 375620224*n + 58236160)*a(n) = 0. (End)
a(n) ~ 25 * 2^(4*n - 3) / (9*Pi*n^2) . - Vaclav Kotesovec, Mar 08 2023

A323847 a(n) = (n-1)(n-2)(n^2+9*n+12)/24.

Original entry on oeis.org

1, 0, 0, 4, 16, 41, 85, 155, 259, 406, 606, 870, 1210, 1639, 2171, 2821, 3605, 4540, 5644, 6936, 8436, 10165, 12145, 14399, 16951, 19826, 23050, 26650, 30654, 35091, 39991, 45385, 51305, 57784, 64856, 72556, 80920, 89985, 99789, 110371, 121771, 134030, 147190

Offset: 0

N. J. A. Sloane, Feb 06 2019

Row 2 of array in A323846.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A323846.

A323847[n_]:=(n-1)(n-2)(n^2+9n+12)/24;Array[A323847,100,0] (* or *)
LinearRecurrence[{5,-10,10,-5,1},{1,0,0,4,16},100] (* Paolo Xausa, Nov 15 2023 *)

G.f.: (x^4-6*x^3+10*x^2-5*x+1)/(1-x)^5. - Alois P. Heinz and Don Knuth, Feb 06 2019

More terms from Alois P. Heinz, Feb 06 2019

Edited with new offset by N. J. A. Sloane, Feb 07 2019 following a suggestion from Don Knuth.

A323967 Number of 3 X n integer matrices (m_{i,j}) such that m_{1,1}=0, m_{3,n}=2, and all rows, columns, and falling diagonals are (weakly) monotonic without jumps of 2.

Original entry on oeis.org

1, 1, 4, 25, 94, 266, 632, 1332, 2570, 4631, 7900, 12883, 20230, 30760, 45488, 65654, 92754, 128573, 175220, 235165, 311278, 406870, 525736, 672200, 851162, 1068147, 1329356, 1641719, 2012950, 2451604, 2967136, 3569962, 4271522, 5084345, 6022116, 7099745

Offset: 0

Alois P. Heinz, Feb 09 2019

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Row (or column) 3 of array in A323846.

a:= n-> `if`(n=0, 1, 2+((((((n+12)*n+55)*n+120)*n-236)*n-312)*n)/360):
seq(a(n), n=0..40);

G.f.: -(x^7-5*x^6+7*x^5+3*x^4-17*x^3+18*x^2-6*x+1)/(x-1)^7.

a(n) = 2+((((((n+12)*n+55)*n+120)*n-236)*n-312)*n)/360 for n > 0, a(0) = 1.

A323968 Number of 4 X n integer matrices (m_{i,j}) such that m_{1,1}=0, m_{4,n}=2, and all rows, columns, and falling diagonals are (weakly) monotonic without jumps of 2.

Original entry on oeis.org

1, 3, 16, 94, 386, 1247, 3423, 8342, 18546, 38304, 74451, 137503, 243103, 413858, 681632, 1090365, 1699493, 2588049, 3859530, 5647620, 8122864, 11500393, 16048805, 22100312, 30062268, 40430198, 53802453, 70896621, 92567829, 119829076, 153873742, 196100423

Offset: 0

Alois P. Heinz, Feb 09 2019

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Row (or column) 4 of array in A323846.

a:= n-> `if`(n=0, 1, 3+((((((((5*n+100)*n+826)*n+4984)*n+15925)*n
          +20020)*n-16756)*n-25104)*n)/40320):
seq(a(n), n=0..35);

G.f.: -(2*x^9-15*x^8+47*x^7-78*x^6+65*x^5-10*x^4-26*x^3+25*x^2-6*x+1)/(x-1)^9.

A323969 Number of 5 X n integer matrices (m_{i,j}) such that m_{1,1}=0, m_{5,n}=2, and all rows, columns, and falling diagonals are (weakly) monotonic without jumps of 2.

Original entry on oeis.org

1, 6, 41, 266, 1247, 4657, 14795, 41586, 106067, 249814, 550334, 1145148, 2268140, 4302757, 7857830, 13873160, 23763590, 39612078, 64424311, 102459670, 159655885, 244167521, 367041525, 543056454, 791755709, 1138709134, 1617041716, 2269272856, 3149514786

Offset: 0

Alois P. Heinz, Feb 09 2019

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Row (or column) 5 of array in A323846.

G.f.: -(3*x^11 -29*x^10 +125*x^9 -314*x^8 +501*x^7 -517*x^6 +323*x^5 -84*x^4 -20*x^3 +30*x^2 -5*x+1) / (x-1)^11.

A323970 Number of 6 X n integer matrices (m_{i,j}) such that m_{1,1}=0, m_{6,n}=2, and all rows, columns, and falling diagonals are (weakly) monotonic without jumps of 2.

Original entry on oeis.org

1, 10, 85, 632, 3423, 14795, 54219, 174844, 508484, 1357051, 3367166, 7846507, 17311702, 36401032, 73344164, 142259423, 266651159, 484610624, 856389171, 1475218962, 2482510921, 4088870385, 6602746625, 10468982846, 16320069070, 25043533065, 37869646820

Offset: 0

Alois P. Heinz, Feb 09 2019

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Row (or column) 6 of array in A323846.

G.f.: -(4*x^13 -47*x^12 +253*x^11 -822*x^10 +1788*x^9 -2728*x^8 +2958*x^7 -2253*x^6 +1145*x^5 -308*x^4 +21*x^3 +33*x^2 -3*x+1) / (x-1)^13.

A323971 Number of 7 X n integer matrices (m_{i,j}) such that m_{1,1}=0, m_{7,n}=2, and all rows, columns, and falling diagonals are (weakly) monotonic without jumps of 2.

Original entry on oeis.org

1, 15, 155, 1332, 8342, 41586, 174844, 642815, 2117690, 6362806, 17671203, 45844681, 112047610, 259796057, 574776968, 1219349012, 2490738686, 4916477305, 9406990883, 17494038498, 31695618318, 56063910644, 96993880940, 164397619093, 273384891666, 446635565576

Offset: 0

Alois P. Heinz, Feb 09 2019

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Row (or column) 7 of array in A323846.

G.f.: -(5*x^15 -69*x^14 +443*x^13 -1750*x^12 +4744*x^11 -9317*x^10 +13630*x^9 -15026*x^8 +12430*x^7-7561*x^6 +3263*x^5-823*x^4 +127*x^3 +35*x^2+1) / (x-1)^15.

A323972 Number of 8 X n integer matrices (m_{i,j}) such that m_{1,1}=0, m_{8,n}=2, and all rows, columns, and falling diagonals are (weakly) monotonic without jumps of 2.

Original entry on oeis.org

1, 21, 259, 2570, 18546, 106067, 508484, 2117690, 7852836, 26400811, 81594028, 234380304, 631352789, 1606571023, 3885713191, 8979237218, 19912769178, 42540796862, 87841523926, 175820917355, 341996038445, 647926774508, 1197980968295, 2165529201795, 3833173915877

Offset: 0

Alois P. Heinz, Feb 09 2019

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Row (or column) 8 of array in A323846.

G.f.: -(6*x^17 -95*x^16 +707*x^15 -3278*x^14 +10588*x^13 -25239*x^12 +45878*x^11 -64775*x^10 +71619*x^9 -62024*x^8 +41650*x^7 -21151*x^6 +7977*x^5 -1820*x^4 +343*x^3 +38*x^2 +4*x+1) / (x-1)^17.

A323973 Number of 9 X n integer matrices (m_{i,j}) such that m_{1,1}=0, m_{9,n}=2, and all rows, columns, and falling diagonals are (weakly) monotonic without jumps of 2.

Original entry on oeis.org

1, 28, 406, 4631, 38304, 249814, 1357051, 6362806, 26400811, 98755951, 337852708, 1069253171, 3159851197, 8786784471, 23140533785, 58032604728, 139238130793, 320911139366, 712976624945, 1531659348507, 3190199728577, 6457722870812, 12731177730290, 24491108899982

Offset: 0

Alois P. Heinz, Feb 09 2019

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Row (or column) 9 of array in A323846.

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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A306322 Number of n X n integer matrices (m_{i,j}) such that m_{1,1}=0, m_{n,n}=2, and all rows, columns, and falling diagonals are (weakly) monotonic without jumps of 2.

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A323847 a(n) = (n-1)*(n-2)*(n^2+9*n+12)/24.

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A323967 Number of 3 X n integer matrices (m_{i,j}) such that m_{1,1}=0, m_{3,n}=2, and all rows, columns, and falling diagonals are (weakly) monotonic without jumps of 2.

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A323968 Number of 4 X n integer matrices (m_{i,j}) such that m_{1,1}=0, m_{4,n}=2, and all rows, columns, and falling diagonals are (weakly) monotonic without jumps of 2.

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A323969 Number of 5 X n integer matrices (m_{i,j}) such that m_{1,1}=0, m_{5,n}=2, and all rows, columns, and falling diagonals are (weakly) monotonic without jumps of 2.

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A323970 Number of 6 X n integer matrices (m_{i,j}) such that m_{1,1}=0, m_{6,n}=2, and all rows, columns, and falling diagonals are (weakly) monotonic without jumps of 2.

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A323971 Number of 7 X n integer matrices (m_{i,j}) such that m_{1,1}=0, m_{7,n}=2, and all rows, columns, and falling diagonals are (weakly) monotonic without jumps of 2.

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A323972 Number of 8 X n integer matrices (m_{i,j}) such that m_{1,1}=0, m_{8,n}=2, and all rows, columns, and falling diagonals are (weakly) monotonic without jumps of 2.

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A323973 Number of 9 X n integer matrices (m_{i,j}) such that m_{1,1}=0, m_{9,n}=2, and all rows, columns, and falling diagonals are (weakly) monotonic without jumps of 2.

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A323847 a(n) = (n-1)(n-2)(n^2+9*n+12)/24.