A223069 -id:A223069 - cp's OEIS frontend

A251801 T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with no 2X2 subblock having x11-x00 less than x10-x01.

150, 1080, 1740, 6627, 28932, 19269, 36552, 348679, 724300, 216912, 187000, 3352272, 17195593, 18577404, 2430631, 905440, 27291608, 294088336, 873945739, 474547100, 27278035, 4206453, 195753488, 3934266515, 26787273856, 44356434596

Offset: 1

R. H. Hardin, Dec 09 2014

Table starts
.......150.........1080............6627.............36552..............187000
......1740........28932..........348679...........3352272............27291608
.....19269.......724300........17195593.........294088336..........3934266515
....216912.....18577404.......873945739.......26787273856........594938582014
...2430631....474547100.....44356434596.....2446811684440......90717637876583
..27278035..12140516092...2255553135793...224175608341756...13902663302194715
.305991368.310512400748.114701510008550.20552060379396424.2134102396738819286

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

R. H. Hardin, Table of n, a(n) for n = 1..112

Column 1 is A184665

Row 1 is A223069(n+1)

Empirical for column k:
k=1: [linear recurrence of order 7]
k=2: [order 20]
Empirical for row n:
n=1: [linear recurrence of order 7]
n=2: [order 22]
n=3: [order 40]
n=4: [order 52]

A250544 T(n,k) = number of (n+1)X(k+1) 0..3 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

150, 1080, 1080, 6627, 10704, 6627, 36552, 79366, 79366, 36552, 187000, 491650, 644779, 491650, 187000, 905440, 2701872, 4169584, 4169584, 2701872, 905440, 4206453, 13657024, 23289547, 27240292, 23289547, 13657024, 4206453, 18933408

Offset: 1

R. H. Hardin, Nov 24 2014

Peter Luschny remarks that the coefficients of the empirical recurrence relation for the column 1 are listed in the 9th row of A246117. - M. F. Hasler, Feb 11 2015

			Some solutions for n=2 k=4
..2..2..1..0..0....2..0..3..1..1....2..2..2..2..0....0..0..2..0..0
..2..3..2..2..2....0..0..3..2..2....2..2..2..2..0....1..1..3..1..1
..1..3..2..3..3....0..0..3..2..3....1..2..2..2..3....0..0..2..0..2
Table starts:
.......150.......1080........6627........36552........187000........905440
......1080......10704.......79366.......491650.......2701872......13657024
......6627......79366......644779......4169584......23289547.....117777788
.....36552.....491650.....4169584.....27240292.....151170400.....752602024
....187000....2701872....23289547....151170400.....824599694....4013192386
....905440...13657024...117777788....752602024....4013192386...19031828420
...4206453...64993652...555362165...3475227442...18064444143...83356429646
..18933408..295871112..2489782728..15210145612...76961701472..345394150196
..83153850.1302924116.10756619019..64036997144..315204675572.1375930596944
.358250280.5595784456.45218540866.262068107390.1254564769204.5328628464360

R. H. Hardin, Table of n, a(n) for n = 1..144

Row/column 1 is A223069(n+1) and row/column 2 of A223071.
Row/column 2-7 are A250538 - A250543; diagonal is A250537.
See also A250676, A250669 - A250675, A250691, A250692...

Empirical for column k (k=2-7 recurrence also works for k=1):
k=1: a(n) = 16*a(n-1) -106*a(n-2) +376*a(n-3) -769*a(n-4) +904*a(n-5) -564*a(n-6) +144*a(n-7)
k=2: [order 16, see A250538]
k=3: [same order 16]
k=4: [same order 16]
k=5: [same order 16]
k=6: [same order 16]
k=7: [same order 16]

Edited by M. F. Hasler, Feb 11 2015

A224123 T(n,k)=Number of nXk 0..3 arrays with row sums nondecreasing and column sums unimodal.

Original entry on oeis.org

4, 16, 10, 50, 150, 20, 130, 1747, 1080, 35, 296, 16782, 47059, 6627, 56, 610, 140210, 1703178, 1070499, 36552, 84, 1163, 1050460, 53355889, 145901795, 21632718, 187000, 120, 2083, 7227405, 1491827492, 17292618579, 11066001160, 400993828

Offset: 1

R. H. Hardin Mar 31 2013

Table starts
...4......16.........50..........130...........296...........610
..10.....150.......1747........16782........140210.......1050460
..20....1080......47059......1703178......53355889....1491827492
..35....6627....1070499....145901795...17292618579.1830213565657
..56...36552...21632718..11066001160.4964366292229
..84..187000..400993828.766386005577
.120..905440.6962188526
.165.4206453
.220

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

R. H. Hardin, Table of n, a(n) for n = 1..49

Column 1 is A000292(n+1)
Column 2 is A223069
Row 1 is A223659

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

A255002 Coefficients of recurrence for rows and columns of A250544 and rows of A250691.

Original entry on oeis.org

-1, 30, -415, 3514, -20386, 85924, -272198, 661180, -1244717, 1822478, -2068955, 1802474, -1181760, 563888, -184752, 37152, -3456

Offset: 0

M. F. Hasler, Feb 11 2015

The convention for signs and indices is such that A(n) = sum_{k=1..16} a(k)*A(n-k), where A is any row of A250691 (i.e., A250692 - A250698) or row or column of A250544 (i.e., A223069, A250538 - A250543).

Cf. A250691, A250692 - A250698; A250544, A223069, A250538 - A250543.

Vecrev(-denominator(ggf(A250538=b2v(readvec("/tmp/b250538.txt"))))) \\ using the "guess generating function" and "b-file to vector" scripts found on the OEIS wiki

cp's OEIS Frontend

A251801 T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with no 2X2 subblock having x11-x00 less than x10-x01.

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A250544 T(n,k) = number of (n+1)X(k+1) 0..3 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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A224123 T(n,k)=Number of nXk 0..3 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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Formula

A255002 Coefficients of recurrence for rows and columns of A250544 and rows of A250691.

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PARI