A002623 -id:A002623 - cp's OEIS frontend

A266428 T(n,k)=Number of nXk binary arrays with rows and columns lexicographically nondecreasing and column sums nondecreasing.

2, 3, 3, 4, 7, 4, 5, 14, 13, 5, 6, 25, 39, 22, 6, 7, 41, 106, 96, 34, 7, 8, 63, 259, 404, 212, 50, 8, 9, 92, 574, 1556, 1391, 433, 70, 9, 10, 129, 1170, 5365, 8764, 4383, 826, 95, 10, 11, 175, 2223, 16585, 49894, 45907, 12758, 1493, 125, 11, 12, 231, 3982, 46463, 251381

Offset: 1

R. H. Hardin, Dec 29 2015

Table starts
..2...3....4......5........6..........7............8............9
..3...7...14.....25.......41.........63...........92..........129
..4..13...39....106......259........574.........1170.........2223
..5..22...96....404.....1556.......5365........16585........46463
..6..34..212...1391.....8764......49894.......251381......1122721
..7..50..433...4383....45907.....448649......3889553.....29520031
..8..70..826..12758...223075....3825307.....59155748....798834778
..9..95.1493..34611..1005991...30555624....861030491..21325003746
.10.125.2575..88206..4224203..227542455..11809616668.546283341439
.11.161.4270.212609.16588684.1579153474.151566391972

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

R. H. Hardin, Table of n, a(n) for n = 1..145

Column 1 and row 1 are A000027(n+1).
Column 2 is A002623.
Row 2 is A004006(n+1).

Empirical for column k:
k=1: a(n) = 2*a(n-1) -a(n-2)
k=2: a(n) = 3*a(n-1) -2*a(n-2) -2*a(n-3) +3*a(n-4) -a(n-5)
k=3: [order 12] Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = (1/6)*n^3 + (1/2)*n^2 + (4/3)*n + 1
n=3: [polynomial of degree 6]
n=4: [polynomial of degree 11]
n=5: [polynomial of degree 19]
n=6: [polynomial of degree 33]
n=7: [polynomial of degree 57]

A267245 T(n,k)=Number of nXk binary arrays with row sums nondecreasing and columns lexicographically nondecreasing.

Original entry on oeis.org

2, 3, 3, 4, 7, 4, 5, 13, 15, 5, 6, 22, 42, 31, 6, 7, 34, 105, 141, 63, 7, 8, 50, 232, 567, 486, 127, 8, 9, 70, 475, 1986, 3351, 1685, 255, 9, 10, 95, 904, 6292, 20040, 20676, 5804, 511, 10, 11, 125, 1632, 18205, 107015, 220235, 129129, 19769, 1023, 11, 12, 161, 2806

Offset: 1

R. H. Hardin, Jan 12 2016

Table starts
..2....3......4........5..........6............7..............8
..3....7.....13.......22.........34...........50.............70
..4...15.....42......105........232..........475............904
..5...31....141......567.......1986.........6292..........18205
..6...63....486.....3351......20040.......107015.........516084
..7..127...1685....20676.....220235......2093467.......17892539
..8..255...5804...129129....2499080.....43555569......683027146
..9..511..19769...804817...28501471....924051709....27044976947
.10.1023..66544..4982759..323067002..19614050515..1079112886476
.11.2047.221581.30629206.3626695952.413556580944.42860145907558

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

R. H. Hardin, Table of n, a(n) for n = 1..216

Column 1 and row 1 are A000027(n+1).
Column 2 is A000225(n+1).
Row 2 is A002623.
Row 3 is A233302(n-1).
Row 4 is A233303(n-1).

Empirical for column k:
k=1: a(n) = 2*a(n-1) -a(n-2)
k=2: a(n) = 3*a(n-1) -2*a(n-2)
k=3: a(n) = 10*a(n-1) -39*a(n-2) +76*a(n-3) -79*a(n-4) +42*a(n-5) -9*a(n-6)
k=4: [order 10]
k=5: [order 14]
k=6: [order 22]
k=7: [order 32]
Empirical for row n:
n=1: a(n) = 2*a(n-1) -a(n-2)
n=2: a(n) = 3*a(n-1) -2*a(n-2) -2*a(n-3) +3*a(n-4) -a(n-5)
n=3: [order 13]

A274537 Number T(n,k) of set partitions of [n] into k blocks such that each element is contained in a block whose index parity coincides with the parity of the element; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 3, 2, 1, 0, 0, 1, 3, 7, 2, 1, 0, 0, 1, 7, 14, 13, 3, 1, 0, 0, 1, 7, 35, 26, 22, 3, 1, 0, 0, 1, 15, 70, 113, 66, 34, 4, 1, 0, 0, 1, 15, 155, 226, 311, 102, 50, 4, 1, 0, 0, 1, 31, 310, 833, 933, 719, 200, 70, 5, 1

Offset: 0

Alois P. Heinz, Jun 27 2016

All odd elements are in blocks with an odd index and all even elements are in blocks with an even index.

			T(6,2) = 1: 135|246.
T(6,3) = 3: 13|246|5, 15|246|3, 1|246|35.
T(6,4) = 7: 13|24|5|6, 15|24|3|6, 1|24|35|6, 15|26|3|4, 15|2|3|46, 1|26|35|4, 1|2|35|46.
T(6,5) = 2: 1|26|3|4|5, 1|2|3|46|5.
T(6,6) = 1: 1|2|3|4|5|6.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 1,  1;
  0, 0, 1,  1,   1;
  0, 0, 1,  3,   2,   1;
  0, 0, 1,  3,   7,   2,   1;
  0, 0, 1,  7,  14,  13,   3,   1;
  0, 0, 1,  7,  35,  26,  22,   3,  1;
  0, 0, 1, 15,  70, 113,  66,  34,  4, 1;
  0, 0, 1, 15, 155, 226, 311, 102, 50, 4, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Partition of a set

Row sums give A274538.
Columns k=0-10 give: A000007, A000007(n-1), A000012(n-2), A052551(n-3), A274868, A274869, A274870, A274871, A274872, A274873, A274874.
T(2n,n) gives A274875.
Main diagonal and lower diagonals give: A000012, A004526, A002623(n-2) or A173196.
Cf. A364267.

b:= proc(n, m, t) option remember; `if`(n=0, x^m, add(
     `if`(irem(j, 2)=t, b(n-1, max(m, j), 1-t), 0), j=1..m+1))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0, 1)):
seq(T(n), n=0..12);

b[n_, m_, t_] := b[n, m, t] = If[n==0, x^m, Sum[If[Mod[j, 2]==t, b[n-1, Max[m, j], 1-t], 0], {j, 1, m+1}]]; T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0, 1]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 18 2016, after Alois P. Heinz *)

Sum_{k=0..n} k * T(n,k) = A364267(n). - Alois P. Heinz, Jul 16 2023

A122432 Riordan array (1/(1+x)^3,x).

Original entry on oeis.org

1, -3, 1, 6, -3, 1, -10, 6, -3, 1, 15, -10, 6, -3, 1, -21, 15, -10, 6, -3, 1, 28, -21, 15, -10, 6, -3, 1, -36, 28, -21, 15, -10, 6, -3, 1, 45, -36, 28, -21, 15, -10, 6, -3, 1, -55, 45, -36, 28, -21, 15, -10

Offset: 0

Paul Barry, Sep 04 2006

Sequence array for (-1)^n*C(n+2,2). Inverse of A122431. Row sums are -A083392(n+1). Antidiagonal sums are (-1)^n*A002623(n).
Call the unsigned version of this array M and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/
having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite matrix product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to A127893. - Peter Bala, Jul 22 2014
From Wolfdieter Lang, Apr 05 2020: (Start)
Triangle T(n, k) has the k=0 column (-1)^n*A000217(n+1) = (-1)^n*binomial(n+2, 2), then repeated and down-shifted.
The unsigned triangle, i.e., Tup(n, k) := (-1)^(n-k)*T(n-1,k-1) = binomial(n-k+2, 2) with n >= 1, k = 1..n, gives the number of triangles of length k (in some units), for k = 1..n, in the matchstick arrangement (or tower of cards, with n cards as basis) with an enclosing triangle of length n, but only triangles with orientation (up) like the enclosing triangle are counted. The total number of matchsticks (cards) is 3*A000217(n). (See the comment by Andrew Howroyd in A085691). Recurrence: Tup(n, k) = 0 for n < k, Tup(1, 1) = 1, and Tup(n, k) = Tup(n-1, k) + n - k + 1, for n >= 2, k = 1..n. Row sums give A000292(n). (End)

			The triangle T(n, k) begins:
n\k  0   1   2   3   4   5   6  7  8  9 ...
-------------------------------------------
0:   1
1  :-3   1
2:   6  -3   1
3: -10   6  -3   1
4:  15 -10   6  -3   1
5; -21  15 -10   6  -3   1
6:  28 -21  15 -10   6  -3   1
7: -36  28 -21  15 -10   6  -3  1
8:  45 -36  28 -21  15 -10   6 -3  1
9: -55  45 -36  28 -21  15 -10  6 -3  1
... reformattet by - _Wolfdieter Lang_, Apr 05 2020

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000217, A000292, A083392, A085691, A122431, A127893.

/* As triangle */ [[(-1)^(n-k)*Binomial(n-k+2, 2): k in [1..n]]: n in [1..10]]; // G. C. Greubel, Oct 29 2017

Table[(-1)^(n - k)*Binomial[n - k + 2, 2], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Oct 29 2017 *)

for(n=0,10, for(k=0,n, print1((-1)^(n-k)*binomial(n-k+2,2), ", "))) \\ G. C. Greubel, Oct 29 2017

Number triangle T(n, k) = [k<=n]*(-1)^(n-k)*binomial(n-k+2, 2).

Recurrence: T(n, k) = - T(n-1, k) + (-1)^(n-k)*(n-k+1), for n >= 0, and k = 0..n. - Wolfdieter Lang, Apr 06 2020

A233301 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays x(i,j) with row sums sum{x(i,j), j=1..k+1} nondecreasing, and column sums sum{i^2*x(i,j), i=1..n+1} nondecreasing.

Original entry on oeis.org

7, 13, 15, 22, 42, 31, 34, 105, 141, 64, 50, 232, 567, 502, 129, 70, 475, 1986, 3556, 1739, 258, 95, 904, 6292, 21957, 21856, 5964, 515, 125, 1632, 18205, 122022, 239330, 135636, 20185, 1029, 161, 2806, 48913, 616439, 2353493, 2694620, 836259, 67609

Offset: 1

R. H. Hardin, Dec 07 2013

Table starts
....7.....13........22.........34..........50..........70..........95
...15.....42.......105........232.........475.........904........1632
...31....141.......567.......1986........6292.......18205.......48913
...64....502......3556......21957......122022......616439.....2871477
..129...1739.....21856.....239330.....2353493....20916337...170084407
..258...5964....135636....2694620....48504411...789640245.11764401320
..515..20185....836259...30257296..1007309118.30406745215
.1029..67609...5134856..338790472.21022231309
.2055.224165..31326263.3761876941
.4107.737347.190404404

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

R. H. Hardin, Table of n, a(n) for n = 1..84

Row 1 is A002623(n+1)

A055609 Number of 3 X n binary matrices with no zero rows or columns, up to row and column permutation.

Original entry on oeis.org

1, 5, 17, 42, 91, 180, 328, 565, 930, 1470, 2248, 3344, 4849, 6881, 9579, 13104, 17649, 23442, 30736, 39833, 51074, 64842, 81574, 101766, 125959, 154771, 188883, 229044, 276085, 330926, 394558, 468083, 552696, 649692, 760482, 886602, 1029691, 1191539, 1374065, 1579326

Offset: 1

Vladeta Jovovic, Jun 03 2000

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3, -1, -3, -1, 3, 6, -6, -3, 1, 3, 1, -3, 1).

Column k=3 of A056152.

Cf. A024206, A002623, A002727.

Vec((G(3, x) - G(2, x))*(1 - x) + O(x^30)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

G.f.: x*(x^8-x^7-x^6-2*x^5+2*x^4+x^3-3*x^2-2*x-1)/((x^3-1)^2*(x^2-1)^2*(x-1)^3).

Terms a(37) and beyond from Andrew Howroyd, Mar 25 2020

A122046 Partial sums of floor(n^2/8).

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 10, 16, 24, 34, 46, 61, 79, 100, 124, 152, 184, 220, 260, 305, 355, 410, 470, 536, 608, 686, 770, 861, 959, 1064, 1176, 1296, 1424, 1560, 1704, 1857, 2019, 2190, 2370, 2560, 2760, 2970, 3190, 3421, 3663, 3916, 4180, 4456, 4744, 5044, 5356, 5681, 6019, 6370

Offset: 0

Roger L. Bagula, Sep 13 2006

Degree of the polynomial P(n+1,x), defined by P(n,x) = [x^(n-1)*P(n-1,x)*P(n-4,x)+P(n-2,x)*P(n-3,x)]/P(n-5,x) with P(1,x)=P(0,x)=P(-1,x)=P(-2,x)=P(-3,x)=1.
Define the sequence b(n) = 1, 4, 10, 20, 36, 60,... for n>=0 with g.f. 1/((1+x)*(1+x^2)*(1-x)^5). Then a(n+3) = b(n)-b(n-1) and b(n)+b(n+1)+b(n+2)+b(n+3) = A052762(n+7)/24. - J. M. Bergot, Aug 21 2013
Maximum Wiener index of all maximal 4-degenerate graphs with n-1 vertices.  (A maximal 4-degenerate graph can be constructed from a 4-clique by iteratively adding a new 4-leaf (vertex of degree 4) adjacent to four existing vertices.)  The extremal graphs are 4th powers of paths, so the bound also applies to 4-trees. - Allan Bickle, Sep 15 2022

			a(6) = 10 = 0 + 0 + 0 + 1 + 2 + 3 + 4.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Allan Bickle and Zhongyuan Che, Wiener indices of maximal k-degenerate graphs, arXiv:1908.09202 [math.CO], 2019.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
A. N. W. Hone, Comments on A122046
A. N. W. Hone, Algebraic curves, integer sequences and a discrete Painlevé transcendent, Proceedings of SIDE 6, Helsinki, Finland, 2004; arXiv:0807.2538 [nlin.SI], 2008. [Set a(n)=d(n+3) on p. 8]
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10a, lambda=4]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,1,-3,3,-1).

Cf. A014125, A052762, A057077, A090294, A122047, A144678, A190718.
Partial sums of A001972.
Maximum Wiener index of all maximal k-degenerate graphs for k=1..6: A000292, A002623, A014125, A122046 (this sequence), A122047, A175724.

[Round((2*n^3+3*n^2-8*n)/48): n in [0..60]]; // Vincenzo Librandi, Jun 25 2011

A122046 := proc(n) round((2*n^3+3*n^2-8*n)/48) ; end proc: # Mircea Merca

p[n_] := p[n] = Cancel[Simplify[ (x^(n - 1)p[n - 1]p[n - 4] + p[n - 2]*p[n - 3])/p[n - 5]]]; p[ -5] = 1;p[ -4] = 1;p[ -3] = 1;p[ -2] = 1;p[ -1] = 1; Table[Exponent[p[n], x], {n, 0, 20}]
Accumulate[Floor[Range[0,60]^2/8]] (* or *) LinearRecurrence[{3,-3,1,1,-3,3,-1},{0,0,0,1,3,6,10},60] (* Harvey P. Dale, Dec 23 2019 *)

a(n)=(2*n^3+3*n^2-8*n+3)\48 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = Sum_{k=0..n} floor(k^2/8).
a(n) = round((2*n^3 + 3*n^2 - 8*n)/48) = round((4*n^3 + 6*n^2 - 16*n - 9)/96) = floor((2*n^3 + 3*n^2 - 8*n + 3)/48) = ceiling((2*n^3 + 3*n^2 - 8*n - 12)/48). - Mircea Merca
a(n) = a(n-8) + (n-4)^2 + n, n > 8. - Mircea Merca
From Andrew Hone, Jul 15 2008: (Start)
a(n+1) = cos((2*n+1)*Pi/4)/(4*sqrt(2)) + (2*n+3)*(2*n^2 + 6*n - 5)/96 + (-1)^n/32.
a(n+1) = A057077(n+1)/8 + A090294(n-1)/32 + (-1)^n/32.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-4) - 3*a(n-5) + 3*a(n-6) - a(n-7). (End)
O.g.f.: x^3 / ( (1+x)*(x^2+1)*(x-1)^4 ). - R. J. Mathar, Jul 15 2008
From Johannes W. Meijer, May 20 2011: (Start)
a(n+3) = A144678(n) + A144678(n-1) + A144678(n-2) + A144678(n-3);
a(n+3) = Sum_{k=0..6} min(6-k+1,k+1)* A190718(n+k-6). (End)
a(n) = (4*n^3 + 6*n^2 - 16*n - 9 - 3*(-1)^n + 12*(-1)^((2*n - 1 + (-1)^n)/4))/96. - Luce ETIENNE, Mar 21 2014
E.g.f.: ((2*x^3 + 9*x^2 - 3*x - 6)*cosh(x) + 6*(cos(x) + sin(x)) + (2*x^3 + 9*x^2 - 3*x - 3)*sinh(x))/48. - Stefano Spezia, Apr 05 2023

Edited by N. J. A. Sloane, Sep 17 2006, Jul 11 2008, Jul 12 2008

More formulas and better name from Mircea Merca, Nov 19 2010

A122047 Degree of the polynomial P(n,x), defined by a Somos-6 type sequence: P(n,x)=(x^(n-1)P(n-1,x)P(n-5,x) + P(n-2,x)*P(n-4,x))/P(n-6,x), initialized with P(n,x)=1 at n<0.

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 22, 31, 42, 55, 70, 88, 109, 133, 160, 190, 224, 262, 304, 350, 400, 455, 515, 580, 650, 725, 806, 893, 986, 1085, 1190, 1302, 1421, 1547, 1680, 1820, 1968, 2124, 2288, 2460, 2640, 2829, 3027

Offset: 0

Roger L. Bagula, Sep 13 2006

nonn

Maximum Wiener index of all maximal 5-degenerate graphs with n vertices. (A maximal 5-degenerate graph can be constructed from a 5-clique by iteratively adding a new 5-leaf (vertex of degree 5) adjacent to 5 existing vertices.) The extremal graphs are 5th powers of paths, so the bound also applies to 5-trees. - Allan Bickle, Sep 15 2022

Allan Bickle and Zhongyuan Che, Wiener indices of maximal k-degenerate graphs, arXiv:1908.09202 [math.CO], 2019.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
A. N. W. Hone, Algebraic curves, integer sequences and a discrete Painlevé transcendent, Proceedings of SIDE 6, Helsinki, Finland, 2004; arXiv:0807.2538 [nlin.SI], 2008. [Set a(n)=d(n+3) on p. 8]
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231 [quant-ph], 2008. [Eq 10a, lambda=5]

Cf. A014125, A122046.

The maximum Wiener index of all maximal k-degenerate graphs for k=1..6 are given in A000292, A002623, A014125, A122046, A122047 (this sequence), A175724, respectively.

p[n_] := p[n] = Cancel[Simplify[(x^(n - 1)p[n - 1]p[n - 5] + p[n - 2]*p[n - 4])/p[n - 6]]];p[ -6] = 1; p[ -5] = 1; p[ -4] = 1; p[ -3] = 1; p[ -2] = 1; p[ -1] = 1; Table[Exponent[p[n], x], {n, 0, 20}]

Conjectures from R. J. Mathar, Jul 15 2008: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-5) - 3*a(n-6) + 3*a(n-7) - a(n-8);
o.g.f.: x^2/((x^4+x^3+x^2+x+1)(x-1)^4). (End)
Conjecture: a(n) = (A000292(n+1) - n - 2 - (-1)^floor((n-1)/5)*A099443(n+1))/5. - R. J. Mathar, Jul 15 2008
a(n+2) = A144679(n) + A144679(n-1) + A144679(n-2) + A144679(n-3) + A144679(n-4). - Johannes W. Meijer, May 20 2011
a(n) = floor((n^3 + 6*n^2 + 5*n)/30). - Allan Bickle, Sep 15 2022

Edited by N. J. A. Sloane, Jul 15 2008

a(22)-a(43) from R. J. Mathar, Jul 15 2008

A233062 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays x(i,j) with row sums sum{x(i,j), j=1..k+1} nondecreasing, and column sums sum{i*x(i,j), i=1..n+1} nondecreasing.

Original entry on oeis.org

7, 13, 16, 22, 47, 33, 34, 130, 161, 66, 50, 319, 723, 564, 132, 70, 755, 2875, 4393, 1933, 265, 95, 1680, 10650, 30692, 26819, 6529, 530, 125, 3673, 36674, 197625, 334580, 162916, 21904, 1052, 161, 7751, 119979, 1180594, 3849923, 3654675, 986252, 72710

Offset: 1

R. H. Hardin, Dec 03 2013

Table starts
....7.....13........22.........34..........50..........70..........95
...16.....47.......130........319.........755........1680........3673
...33....161.......723.......2875.......10650.......36674......119979
...66....564......4393......30692......197625.....1180594.....6637131
..132...1933.....26819.....334580.....3849923....41069900...410815334
..265...6529....162916....3654675....76267521..1476777204.26815913185
..530..21904....986252...39886178..1522090763.53995730752
.1052..72710...5942365..433495407.30425555678
.2092.238992..35694392.4690717033
.4183.781279.213988742

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

R. H. Hardin, Table of n, a(n) for n = 1..84

Row 1 is A002623(n+1)

A236560 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=3, 0<=k<=floor(n/3)^2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 6, 2, 1, 1, 6, 21, 29, 14, 1, 6, 53, 161, 174, 1, 10, 111, 665, 1713, 1549, 608, 107, 11, 1, 1, 10, 201, 1961, 9973, 24267, 29437, 17438, 4756, 459

Offset: 3

Christopher Hunt Gribble, Jan 30 2014

The first 8 rows of T(n,k) are:
.\ k  0     1     2     3     4     5     6     7     8     9
n
   1     1
   1     1
   1     3
   1     3     6     2     1
   1     6    21    29    14
   1     6    53   161   174
   1    10   111   665  1713  1549   608   107    11     1
  1    10   201  1961  9973 24267 29437 17438  4756   459

			T(6,2) = 6 because the number of equivalence classes of ways of placing 2 3 X 3 square tiles in a 6 X 6 square under all symmetry operations of the square is 6. The portrayal of an example from each equivalence class is:
.___________      ___________      ___________
|     |     |    |     |_____|    |     |     |
|  .  |  .  |    |  .  |     |    |  .  |_____|
|_____|_____|    |_____|  .  |    |_____|     |
|           |    |     |_____|    |     |  .  |
|           |    |           |    |     |_____|
|___________|    |___________|    |_____|_____|
.
.___________      ___________      ___________
|     |     |    |_____ _____|    |_____      |
|  .  |     |    |     |     |    |     |_____|
|_____|_____|    |  .  |  .  |    |  .  |     |
|     |     |    |_____|_____|    |_____|  .  |
|     |  .  |    |           |    |     |_____|
|_____|_____|    |___________|    |_____|_____|

Christopher Hunt Gribble, C++ program

Cf. A054252, A236679, A236757, A236800, A236829, A236865, A236915, A236936, A236939.

It appears that:
T(n,0) = 1, n>= 3
T(n,1) = (floor((n-3)/2)+1)*(floor((n-3)/2+2))/2, n >= 3
T(c+2*3,2) = A131474(c+1)*(3-1) + A000217(c+1)*floor(3^2/4) + A014409(c+2), 0 <= c < 3, c even
T(c+2*3,2) = A131474(c+1)*(3-1) + A000217(c+1)*floor((3-1)(3-3)/4) + A014409(c+2), 0 <= c < 3, c odd
T(c+2*3,3) = (c+1)(c+2)/2(2*A002623(c-1)*floor((3-c-1)/2) + A131941(c+1)*floor((3-c)/2)) + S(c+1,3c+2,3), 0 <= c < 3 where
S(c+1,3c+2,3) =
A054252(2,3),  c = 0
A236679(5,3),  c = 1
A236560(8,3),  c = 2

cp's OEIS Frontend

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A267245 T(n,k)=Number of nXk binary arrays with row sums nondecreasing and columns lexicographically nondecreasing.

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Maple

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A122432 Riordan array (1/(1+x)^3,x).

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Magma

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A233301 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays x(i,j) with row sums sum{x(i,j), j=1..k+1} nondecreasing, and column sums sum{i^2*x(i,j), i=1..n+1} nondecreasing.

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A055609 Number of 3 X n binary matrices with no zero rows or columns, up to row and column permutation.

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A122046 Partial sums of floor(n^2/8).

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Extensions

A122047 Degree of the polynomial P(n,x), defined by a Somos-6 type sequence: P(n,x)=(x^(n-1)*P(n-1,x)*P(n-5,x) + P(n-2,x)*P(n-4,x))/P(n-6,x), initialized with P(n,x)=1 at n<0.

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A122047 Degree of the polynomial P(n,x), defined by a Somos-6 type sequence: P(n,x)=(x^(n-1)P(n-1,x)P(n-5,x) + P(n-2,x)*P(n-4,x))/P(n-6,x), initialized with P(n,x)=1 at n<0.