A001752 -id:A001752 - cp's OEIS frontend

A057967 Triangle T(n,k) of numbers of minimal 4-covers of an unlabeled n+4-set that cover k points of that set uniquely (k=4,..,n+4).

1, 3, 1, 10, 5, 2, 30, 21, 11, 3, 83, 75, 49, 18, 5, 208, 231, 177, 84, 30, 6, 495, 636, 554, 318, 143, 42, 9, 1101, 1603, 1540, 1023, 543, 210, 62, 11, 2327, 3737, 3907, 2904, 1759, 822, 311, 82, 15, 4685, 8163, 9153, 7470, 5012, 2706, 1219, 423, 111, 18, 9041

Offset: 0

Vladeta Jovovic, Oct 17 2000

Row sums give A005784.

			[1], [3, 1], [10, 5, 2], [30, 21, 11, 3], [83, 75, 49, 18], ...; there are 5 minimal 4-covers of an unlabeled 6-set that cover 5 points of that set uniquely.

More information

Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963, A057964, A057965(labeled case), A057966, A057968.

T(n, k) = b(n, k)-b(n-1, k); b(n, k) = coefficient of x^k in x^4/24*(Z(S_n; 12 + 4*x, 12 + 4*x^2, ...) + 8*Z(S_n; 3 + x, 3 + x^2, 12 + 4*x^3, 3 + x^4, 3 + x^5, 12 + 4*x^6, ...) + 6*Z(S_n; 6 + 2*x, 12 + 4*x^2, 6 + 2*x^3, 12 + 4*x^4, ...)

+ 3*Z(S_n; 4, 12 + 4*x^2, 4, 12 + 4*x^4, ...) + 6*Z(S_n; 2, 4, 2, 12 + 4*x^4, 2, 4, 2, 12 + 4*x^8, ...)), where Z(S_n; x_1, x_2, ..., x_n) is the cycle index of the symmetric group S_n of degree n.

A057972 Number of 5 X n binary matrices with 3 unit columns up to row and column permutations.

Original entry on oeis.org

3, 31, 252, 1776, 11048, 61106, 303664, 1368844, 5651241, 21559133, 76613440, 255411923, 803771681, 2400633464, 6837010458, 18644075466, 48855805143, 123415815229, 301386128354, 713271875603, 1639572164669, 3667859207856

Offset: 3

Vladeta Jovovic, Oct 21 2000

nonn

A unit column of a binary matrix is a column with only one 1. First differences of a(n) give number of minimal 5 - covers of an unlabeled n - set that cover 8 points of that set uniquely (if offset is 8).

Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963 - A057968, A057969 - A057971.

Number of 5 x n binary matrices with k unit columns up to row and column permutations is coefficient of x^k in (1/5!)*(Z(S_n; 27 + 5*x, 27 + 5*x^2, ...) + 10*Z(S_n; 13 + 3*x, 27 + 5*x^2, 13 + 3*x^3, 27 + 5*x^4, ...) + 15*Z(S_n; 7 + x, 27 + 5*x^2, 7 + x^3, 27 + 5*x^4, ...) + 20*Z(S_n; 6 + 2*x, 6 + 2*x^2, 27 + 5*x^3, 6 + 2*x^4, 6 + 2*x^5, 27 + 5*x^6, ...) + 20*Z(S_n; 4, 6 + 2*x^2, 13 + 3*x^3, 6 + 2*x^4, 4, 27 + 5*x^6, 4, 6 + 2*x^8, 13 + 3*x^9, 6 + 2*x^10, 4, 27 + 5*x^12, ...) + 30*Z(S_n; 3 + x, 7 + x^2, 3 + x^3, 27 + 5*x^4, 3 + x^5, 7 + x^6, 3 + x^7, 27 + 5*x^8, ...) + 24*Z(S_n; 2, 2, 2, 2, 27 + 5*x^5, 2, 2, 2, 2, 27 + 5*x^10, ...)), where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.

G.f. : x^3/120*(35/(1 - x^1)^27 + 130/(1 - x^1)^13/(1 - x^2)^7 + 45/(1 - x^1)^7/(1 - x^2)^10 + 100/(1 - x^1)^6/(1 - x^3)^7 + 20/(1 - x^1)^4/(1 - x^2)^1/(1 - x^3)^3/(1 - x^6)^2 + 30/(1 - x^1)^3/(1 - x^2)^2/(1 - x^4)^5).

A128494 Coefficient table for sums of Chebyshev's S-Polynomials.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, -1, 1, 1, 1, -1, -2, 1, 1, 1, 2, -2, -3, 1, 1, 0, 2, 4, -3, -4, 1, 1, 0, -2, 4, 7, -4, -5, 1, 1, 1, -2, -6, 7, 11, -5, -6, 1, 1, 1, 3, -6, -13, 11, 16, -6, -7, 1, 1, 0, 3, 9, -13, -24, 16, 22, -7, -8, 1, 1, 0, -3, 9, 22, -24, -40, 22, 29, -8, -9, 1, 1, 1, -3, -12, 22, 46, -40, -62, 29, 37, -9, -10, 1, 1, 1, 4, -12

Offset: 0

Wolfdieter Lang, Apr 04 2007

See A049310 for the coefficient table of Chebyshev's S(n,x)=U(n,x/2) polynomials.

This is a 'repetition triangle' based on a signed version of triangle A059260: a(2*p,2*k) = a(2*p+1,2*k) = A059260(p+k,2*k)*(-1)^(p+k) and a(2*p+1,2*k+1) = a(2*p+2,2*k+1) = A059260(p+k+1,2*k+1)*(-1)^(p+k), k >= 0.

			The triangle a(n,m) begins:
  n\m  0   1   2   3   4   5   6   7   8   9  10
   0:  1
   1:  1   1
   2:  0   1   1
   3:  0  -1   1   1
   4:  1  -1  -2   1   1
   5:  1   2  -2  -3   1   1
   6:  0   2   4  -3  -4   1   1
   7:  0  -2   4   7  -4  -5   1   1
   8:  1  -2  -6   7  11  -5  -6   1   1
   9:  1   3  -6 -13  11  16  -6  -7   1   1
  10:  0   3   9 -13 -24  16  22  -7  -8   1   1
... reformatted by _Wolfdieter Lang_, Oct 16 2012
Row polynomial S(1;4,x) = 1 - x - 2*x^2 + x^3 + x^4 = Sum_{k=0..4} S(k,x).
S(4,y)*S(5,y)/y = 3 - 13*y^2 + 16*y^4 - 7*y^6 + y^8, with y=sqrt(2+x) this becomes S(1;4,x).
From _Wolfdieter Lang_, Oct 16 2012: (Start)
S(1;4,x) = (1 - (S(5,x) - S(4,x)))/(2-x) = (1-x)*(2-x)*(1+x)*(1-x-x^2)/(2-x) = (1-x)*(1+x)*(1-x-x^2).
S(5,x) - S(4,x) = R(11,sqrt(2+x))/sqrt(2+x) = -1 + 3*x + 3*x^2 - 4*x^3 - x^4 + x^5. (End)

Wolfdieter Lang, First 15 rows
Per Alexandersson, Luis Angel González-Serrano, and Egor A. Maximenko, Mario Alberto Moctezuma-Salazar, Symmetric polynomials in the symplectic alphabet and their expression via Dickson-Zhukovsky variables, arXiv:1912.12725 [math.CO], 2019.
Per Alexandersson, Luis Angel González-Serrano, Egor A. Maximenko, and Mario Alberto Moctezuma-Salazar, Symmetric Polynomials in the Symplectic Alphabet and the Change of Variables z_j = x_j + x_j^(-1), The Elec. J. of Combinatorics (2021) Vol. 28, No. 1, #P1.56.

Row sums (signed): A021823(n+2). Row sums (unsigned): A070550(n).
Cf. A128495 for S(2; n, x) coefficient table.
The column sequences (unsigned) are, for m=0..4: A021923, A002265, A008642, A128498, A128499.
For m >= 1 the column sequences (without leading zeros) are of the form a(m, 2*k) = a(m, 2*k+1) = ((-1)^k)*b(m, k) with the sequences b(m, k), given for m=1..11 by A008619, A002620, A002623, A001752, A001753, A001769, A001779, A001780, A001781, A001786, A001808.

S(1;n,x) = Sum_{k=0..n} S(k,x) = Sum_{m=0..n} a(n,m)*x^m, n >= 0.
a(n,m) = [x^m](S(n,y)*S(n+1,y)/y) with y:=sqrt(2+x).
G.f. for column m: (x^m)/((1-x)*(1+x^2)^(m+1)), which shows that this is a lower diagonal matrix of the Riordan type, named (1/((1+x^2)*(1-x)), x/(1+x^2)).
From Wolfdieter Lang, Oct 16 2012: (Start)
a(n,m) = [x^m](1- (S(n+1,x) - S(n,x)))/(2-x). From the Binet - de Moivre formula for S and use of the geometric sum.
a(n,m) = [x^m](1- R(2*n+3,sqrt(2+x))/sqrt(2+x))/(2-x) with the monic integer T-polynomials R with coefficient triangle given in A127672. From the odd part of the bisection of the T-polynomials. (End)

A152205 Triangle read by rows, A000012 * A152204.

Original entry on oeis.org

1, 4, 9, 1, 16, 4, 25, 9, 1, 36, 16, 4, 49, 25, 9, 1, 64, 36, 16, 4, 81, 49, 25, 9, 1, 100, 64, 36, 16, 4, 121, 81, 49, 25, 9, 1, 144, 100, 64, 36, 16, 4, 169, 121, 81, 49, 25, 9, 1

Offset: 1

Gary W. Adamson, Nov 29 2008

Row sums = A000292, the tetrahedral numbers.
From Gary W. Adamson, Feb 14 2010: (Start)
Let the triangle = M. Then lim_{n->inf} M^n = A173277 as a left-shifted vector: (1, 4, 13, 32, 74, 152, 298, ...) = A(x), where A(x) satisfies A000290 = A(x)/A(x^2), A000290 = integer squares.
M * [1, 2, 3, ...] = A001752: (1, 4, 11, 24, 46, 80, 130, ...).
M * [1, 3, 6, 10, ...] = A028346: (1, 4, 12, 28, 58, 108, ...). (End)

			First few rows of the triangle:
    1;
    4;
    9,   1;
   16,   4;
   25,   9,   1;
   36,  16,   4;
   49,  25,   9,   1;
   64,  36,  16,   4;
   81,  49,  25,   9,   1;
  100,  64,  36,  16,   4;
  121,  81,  49,  25,   9,   1;
  144, 100,  64,  36,  16,   4;
  169, 121,  81,  49,  25,   9,   1;
  ...

Cf. A152204, A000292.

Cf. A173277, A001752, A028346. - Gary W. Adamson, Feb 14 2010

A000012 * A152204 = partial sums of A152204 by columns.

A175112 First differences of A175111.

Original entry on oeis.org

1, 120, 1442, 6840, 21122, 51000, 105122, 194040, 330242, 528120, 804002, 1176120, 1664642, 2291640, 3081122, 4059000, 5253122, 6693240, 8411042, 10440120, 12816002, 15576120, 18759842, 22408440, 26565122, 31275000, 36585122

Offset: 0

R. J. Mathar, Feb 13 2010

Convolution of the finite sequence 1,116,967,1672,967,116,1 with A001752. Number of points in the standard root system of the D_5 lattice having L_oo norm n.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Cf. A110907, A117216, A175114.

I:=[1, 120, 1442, 6840, 21122, 51000, 105122]; [n le 7 select I[n] else 4*Self(n-1) - 5*Self(n-2) + 5*Self(n-4) - 4*Self(n-5) + Self(n-6): n in [1..40]]; // Vincenzo Librandi, Dec 19 2012

CoefficientList[Series[(116*x + 967*x^2 + 1672*x^3 + 967*x^4 + 116*x^5 + x^6+1)/((1 + x)*(1 - x)^5), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 19 2012 *)
LinearRecurrence[{4,-5,0,5,-4,1},{1,120,1442,6840,21122,51000,105122},30] (* Harvey P. Dale, Sep 12 2023 *)

a(n) = 4*a(n-1) -5*a(n-2) +5*a(n-4) -4*a(n-5) +a(n-6), n>6.
a(n) = ((2*n+1)^5-(2*n-1)^5)/2+(-1)^n, n>0.
G.f.: (116*x+967*x^2+1672*x^3+967*x^4+116*x^5+x^6+1)/((1+x)*(1-x)^5).

A028346 Expansion of 1/((1-x)^4*(1-x^2)^2).

Original entry on oeis.org

1, 4, 12, 28, 58, 108, 188, 308, 483, 728, 1064, 1512, 2100, 2856, 3816, 5016, 6501, 8316, 10516, 13156, 16302, 20020, 24388, 29484, 35399, 42224, 50064, 59024, 69224, 80784, 93840, 108528, 125001, 143412, 163932, 186732, 212002, 239932, 270732, 304612, 341803

Offset: 0

N. J. A. Sloane

Equals triangle A152205 as an infinite lower triangular matrix * the triangular numbers: [1, 3, 6, ...]. - Gary W. Adamson, Feb 14 2010

a(n) is the number of partitions of n into four kinds of parts 1 and two kinds of parts 2. - Joerg Arndt, Mar 09 2016

Index entries for linear recurrences with constant coefficients, signature (4,-4,-4,10,-4,-4,4,-1).

Cf. A152205, A001752 (for the similar series 1/((1-x)^4*(1-x^2))).

[(n+4)*(2*n^4+32*n^3+172*n^2+352*n+15*(-1)^n+225)/960: n in [0..40]]; // Vincenzo Librandi, Feb 14 2016

CoefficientList[Series[1/((1 - x)^4 (1 - x^2)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 14 2016 *)
LinearRecurrence[{4, -4, -4, 10, -4, -4, 4, -1}, {1, 4, 12, 28, 58, 108, 188, 308}, 100] (* G. C. Greubel, Nov 25 2016 *)

Vec(1/((1-x)^4*(1-x^2)^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = (n+4)*(2*n^4 + 32*n^3 + 172*n^2 + 352*n + 15*(-1)^n + 225)/960. - R. J. Mathar, Apr 01 2010
From Antal Pinter, Jan 08 2016: (Start)
a(n) = C(n + 3, 3) + 2*C(n + 1, 3) + 3*C(n - 1, 3) + 4*C(n - 3, 3) + ...
a(n) = Sum_{i = 1..z} i*C(n + 5 - 2*i,3) where z = (2*n + 3 + (-1)^n)/4.
(End)
a(n) = Sum_{i = 0..n} A002624(i). - Antal Pinter, May 05 2016

A057969 5 x n binary matrices without unit columns up to row and column permutations.

Original entry on oeis.org

1, 5, 24, 115, 551, 2542, 11193, 46547, 182164, 670476, 2325506, 7624434, 23716419, 70253721, 198905506, 540079754, 1410786483, 3555443969, 8667153126, 20484365167, 47037898503, 105143200252, 229178029000

Offset: 0

Vladeta Jovovic, Oct 20 2000

nonn

A unit column of a binary matrix is a column with only one 1. First differences of a(n) give number of minimal 5-covers of an unlabeled n-set that cover 5 points of that set uniquely (if offset is 5).

Vladeta Jovovic, The number of minimal covers of an unlabeled n-set that cover k points of that set uniquely
Vladeta Jovovic, Number of binary matrices with fixed number of unit columns up to row and column permutations

Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963-A057968, A057970-A057972.

a(n)=(1/5!)*(Z(S_n; 27, 27, ...) + 10*Z(S_n; 13, 27, 13, 27, ...) + 15*Z(S_n; 7, 27, 7, 27, ...) + 20*Z(S_n; 6, 6, 27, 6, 6, 27, ...) + 20*Z(S_n; 4, 6, 13, 6, 4, 27, 4, 6, 13, 6, 4, 27, ...) + 30*Z(S_n; 3, 7, 3, 27, 3, 7, 3, 27, ...) + 24*Z(S_n; 2, 2, 2, 2, 27, 2, 2, 2, 2, 27, ...)), where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.

G.f. : 1/120*(1/(1 - x^1)^27 + 10/(1 - x^1)^13/(1 - x^2)^7 + 15/(1 - x^1)^7/(1 - x^2)^10 + 20/(1 - x^1)^6/(1 - x^3)^7 + 20/(1 - x^1)^4/(1 - x^2)^1/(1 - x^3)^3/(1 - x^6)^2 + 30/(1 - x^1)^3/(1 - x^2)^2/(1 - x^4)^5 + 24/(1 - x^1)^2/(1 - x^5)^5).

A057970 5 x n binary matrices with 1 unit column up to row and column permutations.

Original entry on oeis.org

1, 8, 54, 333, 1896, 9874, 47164, 207112, 840323, 3168506, 11170331, 37034409, 116095018, 345785753, 982835676, 2676217504, 7005306389, 17681946594, 43153532167, 102080966243, 234565062960, 524594120393, 1143910860870

Offset: 1

Vladeta Jovovic, Oct 21 2000

nonn

A unit column of a binary matrix is a column with only one 1. First differences of a(n) give number of minimal 5 - covers of an unlabeled n - set that cover 6 points of that set uniquely (if offset is 6).

Vladeta Jovovic, Number of minimal covers of an unlabeled n - set that cover k points of that set uniquely
Vladeta Jovovic, Number of binary matrices with fixed number of unit columns up to row and column permutations

Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963 - A057968, A057970 - A057972, A057969, A057971, A057972.

Number of 5 x n binary matrices with k unit columns up to row and column permutations is coefficient of x^k in (1/5!)*(Z(S_n; 27 + 5*x, 27 + 5*x^2, ...) + 10*Z(S_n; 13 + 3*x, 27 + 5*x^2, 13 + 3*x^3, 27 + 5*x^4, ...) + 15*Z(S_n; 7 + x, 27 + 5*x^2, 7 + x^3, 27 + 5*x^4, ...) + 20*Z(S_n; 6 + 2*x, 6 + 2*x^2, 27 + 5*x^3, 6 + 2*x^4, 6 + 2*x^5, 27 + 5*x^6, ...) + 20*Z(S_n; 4, 6 + 2*x^2, 13 + 3*x^3, 6 + 2*x^4, 4, 27 + 5*x^6, 4, 6 + 2*x^8, 13 + 3*x^9, 6 + 2*x^10, 4, 27 + 5*x^12, ...) + 30*Z(S_n; 3 + x, 7 + x^2, 3 + x^3, 27 + 5*x^4, 3 + x^5, 7 + x^6, 3 + x^7, 27 + 5*x^8, ...) + 24*Z(S_n; 2, 2, 2, 2, 27 + 5*x^5, 2, 2, 2, 2, 27 + 5*x^10, ...)), where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.

G.f.: x/120*(5/(1 - x^1)^27 + 30/(1 - x^1)^13/(1 - x^2)^7 + 15/(1 - x^1)^7/(1 - x^2)^10 + 40/(1 - x^1)^6/(1 - x^3)^7 + 30/(1 - x^1)^3/(1 - x^2)^2/(1 - x^4)^5).

A128499 Fifth column (m=4) of triangle A128494.

Original entry on oeis.org

1, 1, -4, -4, 11, 11, -24, -24, 46, 46, -80, -80, 130, 130, -200, -200, 295, 295, -420, -420, 581, 581, -784, -784, 1036, 1036, -1344, -1344, 1716, 1716, -2160, -2160, 2685, 2685, -3300, -3300, 4015, 4015, -4840, -4840, 5786, 5786, -6864, -6864, 8086, 8086, -9464, -9464, 11011, 11011, -12740

Offset: 0

Wolfdieter Lang, Apr 04 2007

Unsigned, this is the repeated sequence A001752.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-5,5,-10,10,-10,10,-5,5,-1,1).

Cf. A128498 (column m=3).

LinearRecurrence[{1,-5,5,-10,10,-10,10,-5,5,-1,1},{1,1,-4,-4,11,11,-24,-24,46,46,-80},60] (* Harvey P. Dale, Aug 26 2023 *)

Vec(-1/((x-1)*(x^2+1)^5) + O(x^100)) \\ Colin Barker, Mar 14 2015

G.f.: -1 / ((x-1)*(x^2+1)^5). - Corrected by Colin Barker, Mar 14 2015
a(2*k) = a(2*k+1) = ((-1)^k)*A001752(n), k>=0.
a(n) = ((2*n^4+44*n^3+334*n^2+1012*n+993)*(-1)^((2*n-1+(-1)^n)/4)+(4*n^3+66*n^2+332*n+495)*(-1)^((6*n-1+(-1)^n)/4)+48)/1536. - Luce ETIENNE, Mar 14 2015

A139672 Convolution of A008619 and A001400.

Original entry on oeis.org

1, 2, 5, 9, 17, 27, 44, 65, 97, 136, 191, 257, 346, 451, 587, 746, 946, 1177, 1461, 1786, 2178, 2623, 3151, 3746, 4443, 5223, 6126, 7131, 8283, 9558, 11007, 12603, 14403, 16377, 18588, 21003, 23692, 26618, 29858, 33372, 37244, 41430, 46022, 50972

Offset: 1

Alford Arnold, Apr 29 2008, May 01 2008

nonn

This is row 21 of a table of values related to Molien series. It is the product of the sequence on row 3 (A008619) with the sequence on row 7 (A001400).
This table may be constructed by moving the rows of table A008284 to prime locations and generating the composite locations by multiplication in a manner similar to the calculation illustrated in the present sequence.
Rows 1 thru 20 and 22 thru 25 are as follows:
A000012 A008619 A000027 A001399 A002620 A001400 A000217 A006918 A000601 A001401 A002623 A001402 A002621 A097701 A000292 A008630 A002624 A008631 A057524 A002622 A008632 A001752 A117485.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2, 1, -3, 0, -1, 2, 2, -1, 0, -3, 1, 2, -1).

a:= proc(n) local m, r; m:= iquo (n, 12, 'r'); r:= r+1; (19+ (145+ (260+ 15* (r+9)*r+ (405+ 90*r+ 216*m) *m) *m) *m) *m/5+ [0, 1, 2, 5, 9, 17, 27, 44, 65, 97, 136, 191][r]+ [0, 16, 37, 77, 128, 208, 307, 447, 616, 840, 1105, 1441][r]*m/2+ [0, 52, 119, 213, 328, 476, 651, 865, 1112, 1404, 1735, 2117][r]*m^2/2 end: seq (a(n), n=1..50); # Alois P. Heinz, Nov 10 2008

CoefficientList[Series[x/((x^2+x+1)(x^2+1)(x+1)^3 (x-1)^6),{x,0,50}],x] (* or *) LinearRecurrence[{2,1,-3,0,-1,2,2,-1,0,-3,1,2,-1},{0,1,2,5,9,17,27,44,65,97,136,191,257},50] (* Harvey P. Dale, Feb 17 2016 *)

G.f.: x/((x^2+x+1)*(x^2+1)*(x+1)^3*(x-1)^6). - Alois P. Heinz, Nov 10 2008

a(n)= -A049347(n)/27 +(2*n+11)*(6*n^4+132*n^3+914*n^2+2068*n+1055)/69120 -(-1)^n*(51/512+n^2/256+11*n/256+A057077(n)/32 ). - R. J. Mathar, Nov 21 2008

More terms from Alois P. Heinz, Nov 10 2008

Corrected A-number in definition. Added formula. - R. J. Mathar, Nov 21 2008

cp's OEIS Frontend

A057967 Triangle T(n,k) of numbers of minimal 4-covers of an unlabeled n+4-set that cover k points of that set uniquely (k=4,..,n+4).

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A057972 Number of 5 X n binary matrices with 3 unit columns up to row and column permutations.

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A128494 Coefficient table for sums of Chebyshev's S-Polynomials.

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A152205 Triangle read by rows, A000012 * A152204.

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A175112 First differences of A175111.

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A028346 Expansion of 1/((1-x)^4*(1-x^2)^2).

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Magma

Mathematica

PARI

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A057969 5 x n binary matrices without unit columns up to row and column permutations.

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A057970 5 x n binary matrices with 1 unit column up to row and column permutations.

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A128499 Fifth column (m=4) of triangle A128494.

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