A054772 -id:A054772 - cp's OEIS frontend

A054252 Triangle T(n,k) of n X n binary matrices with k=0..n^2 ones under action of dihedral group of the square D_4.

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 8, 16, 23, 23, 16, 8, 3, 1, 1, 3, 21, 77, 252, 567, 1051, 1465, 1674, 1465, 1051, 567, 252, 77, 21, 3, 1, 1, 6, 49, 319, 1666, 6814, 22475, 60645, 136080, 256585, 410170, 559014, 652048, 652048, 559014, 410170, 256585, 136080

Offset: 0

Vladeta Jovovic, May 04 2000

From Geoffrey Critzer, Feb 19 2013: (Start)
Cycle indices for n=2,3,4,5 respectively are:
(1/8)(s[1]^4 + 2*s[1]^2*s[2] + 3*s[2]^2 + 2*s[4]).
(1/8)(s[1]^9 + 4*s[1]^3*s[2]^3 + s[1]s[2]^4 + 2*s[1]*s[4]^2).
(1/8)(s[1]^16 + 2*s[1]^4*s[2]^6 + 2*s[4]^4 + 3*s[2]^8).
(1/8)(s[1]^25 + 4*s[1]^5*s[2]^10 + 2*s[1]*s[4]^6 + s[1]*s[2]^12).
(End)
Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X n square under all symmetry operations of the square. - Christopher Hunt Gribble, Feb 17 2014
From Wolfdieter Lang, Oct 03 2016: (Start)
The cycle index G(n) for a square n X n grid with squares coming in two colors with k squares of one color is for the D_4 group (with 8 elements R(90)^j, S R(90)^j, j=0..3)
  (s[1]^(n^2) + s[2]^(n^2/2) +2*s[4]^(n^2/4))/8 + (s[2]^(n^2/2) + s[1]^n*s[2]^((n^2-n)/2))/4 if n is even,
  s[1]*((s[1]^(n^2-1) + s[2]^((n^2-1)/2) + 2*s[4]^((n^2-1)/4))/8) + s[1]^n*s[2]^(n*(n-1)/2)/2 if n is odd.
See the above comment by Geoffrey Critzer for n=2..5.
The figure counting series is c(x) = 1 + x for coloring, say black and white.
Therefore the counting series is C(n,x) = G(n) with substitution s[2^j] = c(x^(2*j)) = 1 + x^(2^j) for j=0,1,2. Row n gives the coefficients of C(n,x) in rising (or falling) order.  This follows from Pólya's counting theorem.  See the Harary-Palmer reference, p. 42, eq. (2.4.6), and eq. (2.2.11) with n=4 on p. 37 for the cycle index of D_4.
(End)

			T(3,2) = 8 because there are 8 nonisomorphic 3 X 3 binary matrices with two ones under action of D_4:
  [0 0 0] [0 0 0] [0 0 0] [0 0 0]
  [0 0 0] [0 0 0] [0 0 1] [0 0 1]
  [0 1 1] [1 0 1] [0 1 0] [1 0 0]
---------------------------------
  [0 0 0] [0 0 0] [0 0 0] [0 0 1]
  [0 1 0] [0 1 0] [1 0 1] [0 0 0]
  [0 0 1] [0 1 0] [0 0 0] [1 0 0]
Triangle T(n,k) begins:
1;
1, 1;
1, 1, 2,  1,  1;
1, 3, 8, 16, 23, 23, 16, 8, 3, 1;

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 42, (2.4.6), p. 37, (2.2.11).

Heinrich Ludwig, Rows n = 0..16, flattened
Index entries for sequences related to groups

Cf. A014409, A019318, A054247 (row sums), A054772.

(* As a triangle *) Prepend[Prepend[Table[CoefficientList[CycleIndexPolynomial[
GraphData[{"Grid", {n, n}}, "AutomorphismGroup"],Table[Subscript[s, i], {i, 1, 4}]] /. Table[Subscript[s, i] -> 1 + x^i, {i, 1, 4}], x], {n, 2, 10}], {1, 1}], {1}] // Grid (* Geoffrey Critzer, Aug 09 2016 *)

def T(n, k):
    if n == 0 or k == 0 or k == n*n:
        return 1
    grid = graphs.Grid2dGraph(n, n)
    m = grid.automorphism_group().cycle_index().expand(2, 'b, w')
    b, w = m.variables()
    return m.coefficient({b: k, w: n*n-k})
[T(n, k) for n in range(6) for k in range(n*n + 1)] # Freddy Barrera, Nov 23 2018

A004652 Expansion of x(1+x^2+x^4)/((1-x)(1-x^2)*(1-x^3)).

Original entry on oeis.org

0, 1, 1, 3, 4, 7, 9, 13, 16, 21, 25, 31, 36, 43, 49, 57, 64, 73, 81, 91, 100, 111, 121, 133, 144, 157, 169, 183, 196, 211, 225, 241, 256, 273, 289, 307, 324, 343, 361, 381, 400, 421, 441, 463, 484, 507, 529, 553, 576, 601, 625, 651, 676, 703, 729, 757, 784, 813

Offset: 0

N. J. A. Sloane

As a Molien series this arises as (1+x^12)/((1-x^4)*(1-x^8)^2).
Starting (1, 3, 4, ...) = row sums of an infinite triangle with alternate columns of (1, 2, 3, ...) and (1, 0, 0, 0, ...). - Gary W. Adamson, May 14 2010
a(n) is also the number of inequivalent (modulo C_4 rotations) square n X n grids with squares coming in two colors and one square has one of the colors. See the formula from A054772. - Wolfdieter Lang, Oct 03 2016
Also the genus of the complete bipartite graph K_{n+2,n+2}. - Eric W. Weisstein, Jan 19 2018

			From _Gary W. Adamson_, May 14 2010: (Start)
First few rows of the generating triangle =
1;
2, 1;
3, 0, 1;
4, 0, 2, 1;
5, 0, 3, 0, 1;
6, 0, 4, 0, 2, 1;
7, 0, 5, 0, 3, 0, 1;
8, 0, 6, 0, 4, 0, 2, 1;
...
Example: a(7) = 13 = (6 + 0 + 4 + 0 + 2 + 1). (End)
x + x^2 + 3*x^3 + 4*x^4 + 7*x^5 + 9*x^6 + 13*x^7 + 16*x^8 + 21*x^9 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
A. R. Calderbank and N. J. A. Sloane, Double circulant codes over Z_4, J. Algeb. Combin., 6 (1997) 119-131 (Abstract, pdf, ps).
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
J. E. Strapasson, S. I. R. Costa, and M. M. S. Alves, On Genus of Circulant Graphs, arXiv:1004.0244 [math.GN], 2010-2016. - _Jonathan Vos Post_, Apr 05 2010
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Graph Genus
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

First differences give A028242. Cf. A035104, A035106.
A002061(n)=a(2*n-1). A035104(n)=a(n+7)-12. A035106(n)=a(n+3)-1.
Cf. A135840, A000290.
Column 1 of A195040. - Omar E. Pol, Sep 28 2011
Cf. A054772, column 2.

a004652 = ceiling . (/ 4) . fromIntegral . (^ 2)
a004652_list = 0 : 1 : zipWith (+) a004652_list [1..]
-- Reinhard Zumkeller, Dec 18 2013

[Ceiling(n^2/4): n in [0..60] ]; // Vincenzo Librandi, Aug 19 2011

with(combstruct):ZL:=[st,{st=Prod(left,right),left=Set(U,card=r),right=Set(U,card=2)}, unlabeled]: subs(r=1,stack): seq(count(subs(r=2,ZL),size=m+3),m=0..57) ; # Zerinvary Lajos, Mar 09 2007

CoefficientList[Series[x (1 - x + x^2)/((1 - x)^2*(1 - x^2)), {x, 0, 57}], x] (* Michael De Vlieger, Oct 03 2016 *)
Table[Ceiling[n^2/4], {n, 0, 20}] (* Eric W. Weisstein, Jan 19 2018 *)
Ceiling[Range[0, 20]^2/4] (* Eric W. Weisstein, Jan 19 2018 *)
LinearRecurrence[{2, 0, -2, 1}, {1, 1, 3, 4}, {0, 20}] (* Eric W. Weisstein, Jan 19 2018 *)

PARI
```
{a(n) = ceil(n^2 / 4)}
```

a(n) = ceiling(n^2/4).
a(-n) = a(n).
G.f.: x * (1 - x + x^2) / ((1 - x)^2 * (1 - x^2)).
a(n) = a(n-1) + a(n-2) - a(n-3) + 1. a(2*n) = n^2, a(2*n-1) = n^2 - n + 1. - Michael Somos, Apr 21 2000
Interleaves square numbers with centered polygonal numbers: a(2*n)=A000290(n), a(2*n+1)=A002061(n+1). - Paul Barry, Mar 13 2003
For n > 1: a(n) is the digit reversal of n in base A008619(n), where a(n) is written in base 10. - Naohiro Nomoto, Mar 15 2004
a(n) = a(n-2) + n - 1. - Paul Barry, Jul 14 2004
Euler transform of length 6 sequence [ 1, 2, 1, 0, 0, -1]. - Michael Somos, Apr 03 2007
Starting (1, 3, 4, 7, 9, 13, ...), row sums of triangle A135840. - Gary W. Adamson, Dec 01 2007
a(n) = (3/8)*(-1)^(n+1) + 5/8 - (3/4)*(n+1) + (1/4)*(n+2)*(n+1). - Richard Choulet, Nov 27 2008
a(n) = n^2/4 - 3*((-1)^n-1)/8. - Omar E. Pol, Sep 28 2011
a(n) = -n + floor( (n+1)(n+3)/4 ). - Wesley Ivan Hurt, Jun 23 2013
a(n) = A054772(n, 1) = A054772(n, n^2-1), n >= 1. - Wolfdieter Lang, Oct 03 2016
E.g.f.: (x*(x + 1)*exp(x) + 3*sinh(x))/4. - Ilya Gutkovskiy, Oct 03 2016
a(n) = binomial(floor((n+3)/2),2) + binomial(floor((n+(-1)^n)/2),2). - Yuchun Ji, Feb 03 2021

A212714 Number of (w,x,y,z) with all terms in {1,...,n} and |w-x| >= w + |y-z|.

Original entry on oeis.org

0, 0, 2, 10, 32, 78, 162, 300, 512, 820, 1250, 1830, 2592, 3570, 4802, 6328, 8192, 10440, 13122, 16290, 20000, 24310, 29282, 34980, 41472, 48828, 57122, 66430, 76832, 88410, 101250, 115440, 131072, 148240, 167042, 187578, 209952

Offset: 0

Clark Kimberling, May 24 2012

For a guide to related sequences, see A211795.

a(n) is also the number of inequivalent (modulo C_4 rotations) square n X n grids with squares coming in two colors and two squares have one of the colors. See the formula from A054772. - Wolfdieter Lang, Oct 03 2016

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. A211795, A054772.

I:=[0,0,2,10,32,78]; [n le 6 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, Aug 02 2013

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[w - x] >= w + Abs[y - z], s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212714 *)
%/2  (* A011864 except for offset *)
LinearRecurrence[{4, -5, 0, 5, -4, 1}, {0, 0, 2, 10, 32, 78}, 40]
CoefficientList[Series[(2 x^2 + 2 x^3 + 2 x^4) / (1 - 4 x + 5 x^2 - 5 x^4 + 4 x^5 - x^6), {x, 0, 80}], x] (* Vincenzo Librandi, Aug 02 2013 *)

a(n) = 4*a(n-1) - 5*a(n-2) + 5*a(n-4) - 4*a(n-5) + a(n-6).
G.f.: (2*x^2 + 2*x^3 + 2*x^4)/(1 - 4*x + 5*x^2 - 5*x^4 + 4*x^5 - x^6).
a(n) = floor(n^4/8). - Wesley Ivan Hurt, Jul 14 2013
a(n) = A054772(n, 2) = A054772(n, n^2-2), n >= 2. - Wolfdieter Lang, Oct 03 2016

A343874 Array read by antidiagonals: T(n,k) is the number of n X n nonnegative integer matrices with sum of elements equal to k, up to rotational symmetry.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 3, 1, 0, 1, 5, 13, 4, 1, 0, 1, 10, 43, 36, 7, 1, 0, 1, 14, 129, 204, 85, 9, 1, 0, 1, 22, 327, 980, 735, 171, 13, 1, 0, 1, 30, 761, 3876, 5145, 2109, 313, 16, 1, 0, 1, 43, 1619, 13596, 29715, 20610, 5213, 528, 21, 1

Offset: 0

Andrew Howroyd, May 06 2021

			Array begins:
=====================================================
n\k | 0  1   2    3     4      5       6        7
----+------------------------------------------------
  0 | 1  0   0    0     0      0       0        0 ...
  1 | 1  1   1    1     1      1       1        1 ...
  2 | 1  1   3    5    10     14      22       30 ...
  3 | 1  3  13   43   129    327     761     1619 ...
  4 | 1  4  36  204   980   3876   13596    42636 ...
  5 | 1  7  85  735  5145  29715  148561   657511 ...
  6 | 1  9 171 2109 20610 164502 1124382  6744582 ...
  7 | 1 13 313 5213 67769 717509 6457529 50732669 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Rows n=0..4 are A000007, A000012, A008610, A054771, A054773.
Columns k=0..1 are A000012, A004652.
Cf. A054772 (binary case), A318795, A343095, A343875.

U(n,s)={(s(1)^(n^2) + s(1)^(n%2)*(2*s(4)^(n^2\4) + s(2)^(n^2\2)))/4}
T(n,k)={polcoef(U(n,i->1/(1-x^i) + O(x*x^k)), k)}

A242279 Number of inequivalent (mod D_4) ways four checkers can be placed on an n X n board.

Original entry on oeis.org

1, 23, 252, 1666, 7509, 26865, 79920, 209096, 491425, 1064575, 2150076, 4104738, 7458437, 13005041, 21857984, 35598880, 56353185, 87019191, 131364700, 194364050, 282314901, 403316353, 567402672, 787201416, 1078078209, 1459020095, 1952782300, 2587048786, 3394568325

Offset: 2

Heinrich Ludwig, May 10 2014

Heinrich Ludwig, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (4,-1,-16,19,20,-45,0,45,-20,-19,16,1,-4,1).

Cf. A054252, A008805, A014409, A082966, A242358, A019318, A054772.

CoefficientList[Series[x^2*(1 + 19*x + 161*x^2 + 697*x^3 + 1446*x^4 + 2070*x^5 + 1422*x^6 + 766*x^7 + 105*x^8 + 31*x^9 + x^10 + x^11) / ((1-x)^9 * (1+x)^5), {x, 0, 20}], x] (* Vaclav Kotesovec, May 10 2014 *)
LinearRecurrence[{4,-1,-16,19,20,-45,0,45,-20,-19,16,1,-4,1},{0,0,1,23,252,1666,7509,26865,79920,209096,491425,1064575,2150076,4104738},40] (* Harvey P. Dale, May 06 2018 *)

a(n) = (n^8 - 6*n^6 + 40*n^4 - 48*n^3 + 16*n^2 + IF(MOD(n, 2) = 1)*(14*n^4 - 48*n^3 + 34*n^2 - 3))/192.
G.f.: x^2*(1 + 19*x + 161*x^2 + 697*x^3 + 1446*x^4 + 2070*x^5 + 1422*x^6 + 766*x^7 + 105*x^8 + 31*x^9 + x^10 + x^11) / ((1-x)^9 * (1+x)^5). - Vaclav Kotesovec, May 10 2014
a(n) = A054772(n, 4), n >= 2. - Wolfdieter Lang, Oct 03 2016

A242358 Number of inequivalent (mod D_4) ways five checkers can be placed on an n X n board.

Original entry on oeis.org

23, 567, 6814, 47358, 239511, 954226, 3207212, 9414828, 24862239, 60136329, 135311658, 286229762, 574460495, 1101240084, 2028333848, 3605765688, 6211552455, 10402472811, 16984387958, 27099325638, 42342870823, 64905898662, 97761436356, 144885584740, 211543443215

Offset: 3

Heinrich Ludwig, May 11 2014

Heinrich Ludwig, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4,-20,40,16,-100,44,110,-110,-44,100,-16,-40,20,4,-5,1).

Cf. A054252, A008805, A014409, A082966, A242279, A054772.

Drop[CoefficientList[Series[x^3*(-23 - 452*x - 4071*x^2 - 16016*x^3 - 40397*x^4 - 59335*x^5 - 61954*x^6 - 38236*x^7 - 17221*x^8 - 3614*x^9 - 623*x^10 + 20*x^11 + x^12 + x^13)/((x-1)^11*(x+1)^6), {x, 0, 20}], x],3] (* Vaclav Kotesovec, May 11 2014 *)

a(n) = (n^10 - 10*n^8 + 35*n^6 + 52*n^5 - 210*n^4 + 140*n^3 - 56*n^2 + 48*n + IF(MOD(n, 2) = 1)*(52*n^5 - 145*n^4 + 140*n^3 - 80*n^2 + 48*n - 15))/960.
G.f.: x^3*(-23 - 452*x - 4071*x^2 - 16016*x^3 - 40397*x^4 - 59335*x^5 - 61954*x^6 - 38236*x^7 - 17221*x^8 - 3614*x^9 - 623*x^10 + 20*x^11 + x^12 + x^13)/((x-1)^11*(x+1)^6). - Vaclav Kotesovec, May 11 2014
a(n) = A054772(n, 5), n >=3. - Wolfdieter Lang, Oct 03 2016

A276454 a(n) = A276452(n) + A276451(n) + A276449(n).

Original entry on oeis.org

1, 2, 22, 464, 13302, 487152, 21475652, 1106550392, 65221981530, 4327577893800, 319187492622012, 25904823495240144, 2294089575287710984, 220132629099295901408, 22751391952803426496488, 2519687900505935894639088, 297684761086123702744203918

Offset: 1

Jason Y.S. Chiu, Hong-Chang Wang, Chiang, Tung-Ying, Sep 03 2016

For a definition and examples of this problem see the comment section of A276449.
The present sequence {a(n)} gives the number of all orbits under C_4 of 2-colored n X n square grids with n squares of one color.
See A054772(n, k) for the table of these total C_4 orbit numbers for 2-colored grids with any number k from {0,1,...,n^2} of squares of one color. - Wolfdieter Lang, Oct 02 2016

			For n = 4 there are A276449(4) = 4 1-orbits, represented by
   + o o +   o + o o   o o + o   o o o o
   o o o o   o o o +   + o o o   o + + o
   o o o o   + o o o   o o o +   o + + o
   + o o +   o o + o   o + o o   o o o o  .
A276451(4) = 12 2-orbits: one of them is
   + o + o   o o o +
   o o o o   + o o o
   o o o o   o o o +
   o + o +   + o o o  ,
and one can take the first one as representative.
A276452(4) = 448 4-orbits: one of them is represented by
   + + + +
   o o o o
   o o o o
   o o o o .
The complete orbit structure for n=4 is 1^4 2^12 4^448, see A276449(4) = 4, A276451(4) = 12, A276452(4) = 448.
a(4) = 448 + 12 + 4 = 464.
A014062(4) = 448*4 + 12*2 + 4*1 = 1820.

Hong-Chang Wang, Table of n, a(n) for n = 1..70

Cf. A014062, A054772, A276451, A276452, A276454.

f[n_] := If[MemberQ[{2, 3}, #], 0, Function[i, Binomial[(2 i) (2 i + #), i]]@ Floor[n/4]] &@ Mod[n, 4];g[n_] := (Function[j, Binomial[2 j (j + Boole@ OddQ@ n), j]]@ Floor[n/2] - f@ n)/2; Table[(Binomial[n^2, n] - 2 g@ n - f@ n)/4 + (Function[j, Binomial[2 j (j + Boole@ OddQ@ n), j]]@ Floor[n/2] - f@ n)/2 + f@ n, {n, 17}] (* Michael De Vlieger, Sep 12 2016 *)

from math import comb as binomial
for j in range(1, 20):
    t = binomial(j * j, j)
    i = j // 2
    if j % 2 == 0:
        d = binomial(2 * i * i, i)
    else:
        d = binomial(2 * i * (i + 1), i)
    a = (t - d) // 4
    if j % 4 == 0:
        c = binomial((j * j // 4), (j // 4))
    elif j % 4 == 1:
        c = binomial(((j - 1) // 2) * ((j - 1) // 2 + 1), ((j - 1) // 4))
    else:
        c = 0
    b = (d - c) // 2
    print(str(j) + " " + str(a + b + c))

a(n) = A276452(n) + A276451(n) + A276449(n) for n = 1, 2, 3, ...,
A014062(n) = A276452(n)*4 + A276451(n)*2 + A276449(n).
a(n) = A054772(n, 2), n >= 1. - Wolfdieter Lang, Oct 02 2016

Edited by Wolfdieter Lang, Oct 02 2016

A275799 Number of inequivalent (modulo C_4 rotations) square n X n grids with squares coming in two colors and three squares have one of the colors.

Original entry on oeis.org

1, 22, 140, 578, 1785, 4612, 10416, 21340, 40425, 72010, 121836, 197582, 308945, 468328, 690880, 995352, 1404081, 1944030, 2646700, 3549370, 4694921, 6133292, 7921200, 10123828, 12814425, 16076242, 20001996, 24696070, 30273825, 36864080

Offset: 2

Wolfdieter Lang, Oct 03 2016

See the k=3 column of table A054772(n, k), with more explanations there.

Colin Barker, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3,-8,14,0,-14,8,3,-4,1).

Cf. A054772, A000012 (k=0), A004652 (k=1), A212714 (k=2).

Vec(x^2*(1+18*x+55*x^2+92*x^3+55*x^4+18*x^5+x^6)/((1-x)^7*(1+x)^3) + O(x^40)) \\ Colin Barker, Oct 16 2016

a(n) = A054772(n, 3) = A054772(n, n^2-3), n >= 2.
From Colin Barker, Oct 09 2016: (Start)
G.f.: x^2*(1+18*x+55*x^2+92*x^3+55*x^4+18*x^5+x^6) / ((1-x)^7*(1+x)^3).
a(n) = (n^6-3*n^4+2*n^2)/24 for n even.
a(n) = (n^6-3*n^4+5*n^2-3)/24 for n odd. (End)
From Stefan Hollos, Oct 16 2016: (Start)
a(n) = C(n^2,3)/4 for n even,
a(n) = (C(n^2,3) + (n^2-1)/2)/4 for n odd. (End)

A277226 Number of inequivalent (modulo C_4 rotations) square n X n grids with squares coming in two colors and four squares have one of the colors.

Original entry on oeis.org

1, 34, 464, 3182, 14769, 53044, 158976, 416140, 980625, 2124310, 4295376, 8199674, 14907809, 25992232, 43700224, 71167704, 112680801, 173990730, 262690000, 388656070, 564571601, 806527964, 1134722304, 1574255332, 2156041329, 2917838014, 3905408976, 5173826770, 6788930625

Offset: 2

Wolfdieter Lang, Oct 06 2016

See the k=4 column of table A054772(n, k), with more explanations there.

Colin Barker, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,2,27,-36,0,36,-27,-2,12,-6,1).

Cf. A054772, A000012 (k=0), A004652 (k=1), A212714 (k=2), A275799 (k=3).

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x^2*(1+28*x+272*x^2+804*x^3+1150*x^4+804*x^5 +272*x^6+28*x^7+x^8)/((1-x)^9*(1+x)^3))); // G. C. Greubel, Oct 22 2018

CoefficientList[Series[x^2*(1+28*x+272*x^2+804*x^3+1150*x^4+804*x^5 +272*x^6+28*x^7+x^8)/((1-x)^9*(1+x)^3), {x, 0, 50}], x] (* G. C. Greubel, Oct 22 2018 *)

Vec(x^2*(1+28*x+272*x^2+804*x^3+1150*x^4+804*x^5+272*x^6+28*x^7 +x^8)/((1-x)^9*(1+x)^3) + O(x^40)) \\ Colin Barker, Oct 16 2016

a(n) = A054772(n, 4) = A054772(n, n^2-4), n >= 2.
From Colin Barker, Oct 09 2016: (Start)
G.f.: x^2*(1+28*x+272*x^2+804*x^3+1150*x^4+804*x^5+272*x^6+28*x^7+x^8) / ((1-x)^9*(1+x)^3).
a(n) = (n^8-6*n^6+14*n^4)/96 for n even.
a(n) = (n^8-6*n^6+14*n^4-6*n^2-3)/96 for n odd. (End)
From Stefan Hollos, Oct 16 2016: (Start)
a(n) = (C(n^2,4) + C(n^2/2,2) + n^2/2)/4 for n even,
a(n) = (C(n^2,4) + C((n^2-1)/2,2) + (n^2-1)/2)/4 for n odd. (End)

cp's OEIS Frontend

A054252 Triangle T(n,k) of n X n binary matrices with k=0..n^2 ones under action of dihedral group of the square D_4.

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A004652 Expansion of x*(1+x^2+x^4)/((1-x)*(1-x^2)*(1-x^3)).

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A212714 Number of (w,x,y,z) with all terms in {1,...,n} and |w-x| >= w + |y-z|.

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A343874 Array read by antidiagonals: T(n,k) is the number of n X n nonnegative integer matrices with sum of elements equal to k, up to rotational symmetry.

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A242279 Number of inequivalent (mod D_4) ways four checkers can be placed on an n X n board.

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A242358 Number of inequivalent (mod D_4) ways five checkers can be placed on an n X n board.

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A276454 a(n) = A276452(n) + A276451(n) + A276449(n).

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A275799 Number of inequivalent (modulo C_4 rotations) square n X n grids with squares coming in two colors and three squares have one of the colors.

A004652 Expansion of x(1+x^2+x^4)/((1-x)(1-x^2)*(1-x^3)).