A222993 -id:A222993 - cp's OEIS frontend

A251065 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with no 2X2 subblock having x11-x00 less than x10-x01.

50, 222, 336, 867, 2740, 2167, 3123, 17129, 31895, 14180, 10660, 89429, 320950, 379440, 92429, 35064, 410718, 2482824, 6182747, 4499219, 603249, 112373, 1716324, 15857764, 71431719, 119032248, 53420941, 3935721, 353517, 6678657, 87634164

Offset: 1

R. H. Hardin, Nov 29 2014

Table starts
......50.......222.........867..........3123..........10660...........35064
.....336......2740.......17129.........89429.........410718.........1716324
....2167.....31895......320950.......2482824.......15857764........87634164
...14180....379440.....6182747......71431719......641508138......4755605792
...92429...4499219...119032248....2062530658....26192388204....262314968432
..603249..53420941..2295692500...59730054958..1074828860950..14586457826568
.3935721.634161915.44280784440.1731036678270.44184817939580.813777827212550

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

R. H. Hardin, Table of n, a(n) for n = 1..180

Column 1 is A184556

Row 1 is A222993(n+1)

Empirical for column k:
k=1: a(n) = 6*a(n-1) +7*a(n-2) -23*a(n-3) -4*a(n-4) +12*a(n-5)
k=2: [order 11]
k=3: [order 39]
Empirical for row n:
n=1: a(n) = 9*a(n-1) -31*a(n-2) +51*a(n-3) -40*a(n-4) +12*a(n-5)
n=2: [order 12]
n=3: [order 21]
n=4: [order 27]
n=5: [order 33]
n=6: [order 39]

A250527 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nondecreasing x(i,j)-x(i-1,j) in the j direction.

Original entry on oeis.org

50, 222, 222, 867, 1180, 867, 3123, 5029, 5029, 3123, 10660, 18859, 21955, 18859, 10660, 35064, 65310, 82023, 82023, 65310, 35064, 112373, 214812, 279161, 300131, 279161, 214812, 112373, 353517, 682921, 896191, 993123, 993123, 896191, 682921

Offset: 1

R. H. Hardin, Nov 24 2014

Table starts
......50......222......867......3123.....10660......35064.....112373.....353517
.....222.....1180.....5029.....18859.....65310.....214812.....682921....2122743
.....867.....5029....21955.....82023....279161.....896191....2771901....8374485
....3123....18859....82023....300131....993123....3088923....9240559...26984403
...10660....65310...279161....993123...3183434....9580060...27710543...78195145
...35064...214812...896191...3088923...9580060...27910024...78204775..213775147
..112373...682921..2771901...9240559..27710543...78204775..212707851..565044857
..353517..2122743..8374485..26984403..78195145..213775147..565044857.1462376991
.1097430..6501118.24944039..77707851.217483704..575554760.1478308633.3731976469
.3374226.19720580.73714737.222271083.600720730.1537251580.3832617341.9435234815

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

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

Column 1 is A222993(n+1)

Empirical for column k (k=2 recurrence also works for k=1):
k=1: a(n) = 9*a(n-1) -31*a(n-2) +51*a(n-3) -40*a(n-4) +12*a(n-5)
k=2-7: a(n) = 14*a(n-1) -85*a(n-2) +294*a(n-3) -639*a(n-4) +906*a(n-5) -839*a(n-6) +490*a(n-7) -164*a(n-8) +24*a(n-9)

A223815 T(n,k)=Number of nXk 0..2 arrays with row sums nondecreasing and column sums unimodal.

Original entry on oeis.org

3, 9, 6, 22, 50, 10, 46, 337, 222, 15, 86, 1922, 4120, 867, 21, 148, 9783, 65465, 43941, 3123, 28, 239, 45537, 921010, 1941904, 426527, 10660, 36, 367, 198252, 11789302, 76453838, 52310070, 3875213, 35064, 45, 541, 817482, 139965348, 2740448352

Offset: 1

R. H. Hardin Mar 27 2013

Table starts
..3.......9.........22...........46.............86.............148
..6......50........337.........1922...........9783...........45537
.10.....222.......4120........65465.........921010........11789302
.15.....867......43941......1941904.......76453838......2740448352
.21....3123.....426527.....52310070.....5767953144....581679349302
.28...10660....3875213...1313561156...405333095737.115166445136785
.36...35064...33540705..31302500115.26977674423993
.45..112373..279734339.716436867795
.55..353517.2266708593
.66.1097430

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

R. H. Hardin, Table of n, a(n) for n = 1..70

Column 1 is A000217(n+1)
Column 2 is A222993
Row 1 is A223718

A287532 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals upwards, where A(n,k) = sum of unimodal products of length n and bound k.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 11, 9, 1, 1, 26, 50, 16, 1, 1, 57, 222, 150, 25, 1, 1, 120, 867, 1080, 355, 36, 1, 1, 247, 3123, 6627, 3775, 721, 49, 1, 1, 502, 10660, 36552, 33502, 10626, 1316, 64, 1, 1, 1013, 35064, 187000, 262570, 128758, 25676, 2220, 81, 1

Offset: 0

Don Knuth, May 26 2017

A unimodal product of length n and parameter k is a product of positive integers a_1 ... a_m ... a_n where a_1 <= ... <= a_m <= k and k >= a_m >= ... >= a_n; furthermore we consider each choice of m to give a distinct product, unless a_m=k. (See the example.)

			A(2,3)=50 because of the products 1*1,1*1,1*1 [m=0,1,2]; 1*2,1*2 [m=1,2]; 1*3; 2*1,2*1 [m=0,1]; 2*2,2*2,2*2 [m=0,1,2]; 2*3; 3*1; 3*2; 3*3; total 50.
Square array begins:
  n\k| 1,   2,    3,     4,      5,       6, ...
  ---+------------------------------------------
   0 | 1,   1,    1,     1,      1,       1, ...
   1 | 1,   4,    9,    16,     25,      36, ...
   2 | 1,  11,   50,   150,    355,     721, ...
   3 | 1,  26,  222,  1080,   3775,   10626, ...
   4 | 1,  57,  867,  6627,  33502,  128758, ...
   5 | 1, 120, 3123, 36552, 262570, 1360128, ...
  ...

A(n,n) gives A383883.
Columns k=5..6 give A383892, A383893.
Cf. A000290, A000295, A222993, A223069.

f[k_]:=Product[1-j x,{j,k}]; A[n_,k_]:=Coefficient[Series[1/f[k]/f[k-1],{x,0,n}],x,n]

a(n, k) = sum(j=0, n, stirling(j+k-1, k-1, 2)*stirling(n-j+k, k, 2)); \\ Seiichi Manyama, May 14 2025

A(n,k) is the coefficient of x^n in 1/((1-k*x) * (1-(k-1)*x)^2 * ... * (1-x)^2).

A(n,k) = Sum_{j=0..n} Stirling2(j+k-1,k-1) * Stirling2(n-j+k,k) for k >= 1. - Seiichi Manyama, May 14 2025

A353047 Number of length n words on alphabet {0,1,2} that contain each of the subwords 01, 02, 10, 12, 20, and 21 as (not necessarily contiguous) subwords.

Original entry on oeis.org

12, 108, 600, 2664, 10404, 37476, 127920, 420768, 1348476, 4242204, 13169160, 40490712, 123635028, 375623892, 1137095520, 3433306896, 10347106860, 31141984140, 93639862200, 281372571720, 845074016772, 2537235316548, 7615933808400, 22856659795584, 68588501433564

Offset: 5

Geoffrey Critzer, Apr 19 2022

Let A be an alphabet containing m letters. Let S be the set of m^2-m ordered two-tuples of distinct letters in A. The generating function for the number of length n words on A that contain each two-tuple in S as a (not necessarily contiguous) subword is m*(m-1)!^2*x^(2*m-1)/((1-m*x)*Product_{k=1..m-1} (1-k*x)^2).

Appears to equal 12 times A222993, except that sequence only has a conjectured formula. - N. J. A. Sloane, Jun 17 2022

			a(5) = 12 because we have: {0, 1, 2, 0, 1}, {0, 1, 2, 1, 0}, {0, 2, 1, 0, 2}, {0, 2, 1, 2, 0}, {1, 0, 2, 0, 1}, {1, 0, 2, 1, 0}, {1, 2, 0, 1, 2}, {1, 2, 0, 2, 1}, {2, 0, 1, 0, 2}, {2, 0, 1, 2, 0}, {2, 1, 0, 1, 2}, {2, 1, 0, 2, 1}.

Index entries for linear recurrences with constant coefficients, signature (9,-31,51,-40,12).

Cf. A058809, A222993, A005803 (binary words).

nn = 15; vertices = Level[Outer[ List, {a, b, c}, {d, e, f}, {h, i, j}, {k, l, m}, {n, o, p}, {q, r, s}], {6}]; x = {a -> b, d -> e, i -> j, o -> p}; y = {b -> c, h -> i, k -> l, r -> s}; z = {e -> f, l -> m, n -> o, q -> r}; replacementlist = Table[vertices[[kk]] -> kk, {kk, 1, 729}]; G= Normal[SparseArray[Flatten[Table[Normal[Merge[ Map[{mm, vertices[[mm]] /. # /. replacementlist} -> 1 &, {x, y, z}], Total]], {mm, 1, 729}]]]]; Iwg =
Inverse[IdentityMatrix[729] - w G]; CoefficientList[ Series[Iwg[[1, 729]], {w, 0, nn}], w]

G.f.: (12*x^5)/((1 - 2*x)^2*(1 - x)^2*(1 - 3*x)).

cp's OEIS Frontend

A251065 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with no 2X2 subblock having x11-x00 less than x10-x01.

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A250527 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nondecreasing x(i,j)-x(i-1,j) in the j direction.

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A223815 T(n,k)=Number of nXk 0..2 arrays with row sums nondecreasing and column sums unimodal.

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A287532 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals upwards, where A(n,k) = sum of unimodal products of length n and bound k.

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A353047 Number of length n words on alphabet {0,1,2} that contain each of the subwords 01, 02, 10, 12, 20, and 21 as (not necessarily contiguous) subwords.

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