A006382 -id:A006382 - cp's OEIS frontend

A005232 Expansion of (1-x+x^2)/((1-x)^2(1-x^2)(1-x^4)).

1, 1, 3, 4, 8, 10, 16, 20, 29, 35, 47, 56, 72, 84, 104, 120, 145, 165, 195, 220, 256, 286, 328, 364, 413, 455, 511, 560, 624, 680, 752, 816, 897, 969, 1059, 1140, 1240, 1330, 1440, 1540, 1661, 1771, 1903, 2024, 2168, 2300, 2456, 2600, 2769, 2925, 3107, 3276

Offset: 0

N. J. A. Sloane

Also number of n X 2 binary matrices under row and column permutations and column complementations (if offset is 0).
Also Molien series for certain 4-D representation of dihedral group of order 8.
With offset 4, number of bracelets (turnover necklaces) of n-bead of 2 colors with 4 red beads. - Washington Bomfim, Aug 27 2008
From Vladimir Shevelev, Apr 23 2011: (Start)
Also number of non-equivalent necklaces of 4 beads each of them painted by one of n colors.
The sequence solves the so-called Reis problem about convex k-gons in case k=4 (see our comment to A032279). (End)
Number of 2 X 2 matrices with nonnegative integer values totaling n under row and column permutations. - Gabriel Burns, Nov 08 2016
From Petros Hadjicostas, Jan 12 2019: (Start)
By "necklace", Vladimir Shevelev (above) means "turnover necklace", i.e., a bracelet. Zagaglia Salvi (1999) also uses this terminology: she calls a bracelet "necklace" and a necklace "cycle".
According to Cyvin et al. (1997), the sequence (a(n): n >= 0) consists of "the total numbers of isomers of polycyclic conjugated hydrocarbons with q + 1 rings and q internal carbons in one ring (class Q_q)", where q = 4 and n is the hydrogen content (i.e., we count certain isomers of C_{n+2*q} H_n with q = 4 and n >= 0). (End)

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 8*x^4 + 10*x^5 + 16*x^6 + 20*x^7 + 29*x^8 + ...
There are 8 4 X 2 matrices up to row and column permutations and column complementations:
  [1 1] [1 0] [1 0] [0 1] [0 1] [0 1] [0 1] [0 0]
  [1 1] [1 1] [1 0] [1 0] [1 0] [1 0] [0 1] [0 1]
  [1 1] [1 1] [1 1] [1 1] [1 0] [1 0] [1 0] [1 0]
  [1 1] [1 1] [1 1] [1 1] [1 1] [1 0] [1 0] [1 1].
There are 8 2 X 2 matrices of nonnegative integers totaling 4 up to row and column permutations:
  [4 0] [3 1] [2 2] [2 1] [2 1] [3 0] [2 0] [1 1]
  [0 0] [0 0] [0 0] [0 1] [1 0] [1 0] [2 0] [1 1].

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. Zagaglia Salvi, Ordered partitions and colourings of cycles and necklaces, Bull. Inst. Combin. Appl., 27 (1999), 37-40.

T. D. Noe, Table of n, a(n) for n = 0..1000
Nesrine Benyahia-Tani, Zahra Yahi, and Sadek Bouroubi Ordered and non-ordered non-congruent convex quadrilaterals inscribed in a regular n-gon. Rostocker Math. Kolloq. 68, 71-79 (2013), Theorem 1.
S. J. Cyvin, B. N. Cyvin, J. Brunvoll, I. Gutman, Chen Rong-si, S. El-Basil, and Zhang Fuji, Polygonal systems including the corannulene and coronene homologs: novel applications of Pólya's theorem, Z. Naturforsch., 52a (1997), 867-873.
S. N. Ethier and S. E. Hodge, Identity-by-descent analysis of sibship configurations, Amer. J. Medical Genetics, 22 (1985), 263-272.
H. Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10 (1979), no. 8, 964-999.
W. D. Hoskins and Anne Penfold Street, Twills on a given number of harnesses, J. Austral. Math. Soc. Ser. A 33 (1982), no. 1, 1-15.
W. D. Hoskins and A. P. Street, Twills on a given number of harnesses, J. Austral. Math. Soc. (Series A), 33 (1982), 1-15. (Annotated scanned copy)
M. Klemm, Selbstduale Codes über dem Ring der ganzen Zahlen modulo 4, Arch. Math. (Basel), 53 (1989), 201-207.
P. Lisonek, Quasi-polynomials: A case study in experimental combinatorics, RISC-Linz Report Series No. 93-18, 1983. (Annotated scanned copy)
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Vladimir Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638.
Vladimir Shevelev, A problem of enumeration of two-color bracelets with several variations, arXiv:0710.1370 [math.CO], 2007-2011.
Vladimir Shevelev, Spectrum of permanent's values and its extremal magnitudes in Lambda_n^3 and Lambda_n(alpha,beta,gamma) (Cf. Section 5), arXiv:1104.4051 [math.CO], 2011.
Index entries for sequences related to bracelets
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-2,0,2,-1).
Index entries for two-way infinite sequences

Row n=2 of A343875.
Column k=4 of A052307.
Cf. A006381, A006382, A008805.

A005232:=-(-1-z-2*z**3+2*z**2+z**7-2*z**6+2*z**4)/(z**2+1)/(1+z)**2/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence apart from an initial 1

k = 4; Table[(Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n + Binomial[If[OddQ[n], n - 1, n - If[OddQ[k], 2, 0]]/2, If[OddQ[k], k - 1, k]/2])/2, {n, k, 50}] (* Robert A. Russell, Sep 27 2004 *)
CoefficientList[ Series[(1 - x + x^2)/((1 - x)^2(1 - x^2)(1 - x^4)), {x, 0, 51}], x] (* Robert G. Wilson v, Mar 29 2006 *)
LinearRecurrence[{2,0,-2,2,-2,0,2,-1},{1,1,3,4,8,10,16,20},60] (* Harvey P. Dale, Oct 24 2012 *)
k=4 (* Number of red beads in bracelet problem *); CoefficientList[Series[(1/k Plus@@(EulerPhi[#] (1-x^#)^(-(k/#))&/@Divisors[k])+(1+x)/(1-x^2)^Floor[(k+2)/2])/2,{x,0,50}],x] (* Herbert Kociemba, Nov 04 2016 *)

{a(n) = (n^3 + 9*n^2 + (32-9*(n%2))*n + [48, 15, 36, 15][n%4+1]) / 48}; \\ Michael Somos, Feb 01 2007

{a(n) = my(s=1); if( n<-5, n = -6-n; s=-1); if( n<0, 0, s * polcoeff( (1 - x + x^2) / ((1 - x)^2 * (1 - x^2) * (1 - x^4)) + x * O(x^n), n))}; \\ Michael Somos, Feb 01 2007

a(n) = round((n^3 +9*n^2 +(32-9*(n%2))*n)/48 +0.6) \\ Washington Bomfim, Jul 17 2008

a(n) = ceil((n+1)*(2*n^2+16*n+39+9*(-1)^n)/96) \\ Tani Akinari, Aug 23 2013

a=lambda n: sum((k//2+1)*((n-k)//2+1) for k in range((n-1)//2+1))+(n+1)%2*(((n//4+1)*(n//4+2))//2)  # Gabriel Burns, Nov 08 2016

G.f.: (1+x^3)/((1-x)*(1-x^2)^2*(1-x^4)).
G.f.: (1/8)*(1/(1-x)^4+3/(1-x^2)^2+2/(1-x)^2/(1-x^2)+2/(1-x^4)). - Vladeta Jovovic, Aug 05 2000
Euler transform of length 6 sequence [ 1, 2, 1, 1, 0, -1 ]. - Michael Somos, Feb 01 2007
a(2n+1) = A006918(2n+2)/2;
a(2n) = (A006918(2n+1) + A008619(n))/2.
a(n) = -a(-6 - n) for all n in Z. - Michael Somos, Feb 05 2011
From Vladimir Shevelev, Apr 22 2011: (Start)
if n == 0 (mod 4), then a(n) = n*(n^2-3*n+8)/48;
if n == 1, 3 (mod 4), then a(n) = (n^2-1)*(n-3)/48;
if n == 2 (mod 4), then a(n) = (n-2)*(n^2-n+6)/48. (End)
a(n) = 2*a(n-1) - 2*a(n-3) + 2*a(n-4) - 2*a(n-5) + 2*a(n-7) - a(n-8) with a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 4, a(4) = 8, a(5) = 10, a(6) = 16, a(7) = 20. - Harvey P. Dale, Oct 24 2012
a(n) = ((n+3)*(2*n^2+12*n+19+9*(-1)^n) + 6*(-1)^((2*n-1+(-1)^n)/4)*(1+(-1)^n))/96. - Luce ETIENNE, Mar 16 2015
a(n) = |A128498(n)| + |A128498(n-3)|. - R. J. Mathar, Jun 11 2019

Sequence extended by Christian G. Bower

A006381 Number of n X 3 binary matrices under row and column permutations and column complementations.

Original entry on oeis.org

1, 1, 4, 7, 19, 32, 68, 114, 210, 336, 562, 862, 1349, 1987, 2950, 4201, 5991, 8278, 11422, 15386, 20660, 27218, 35718, 46158, 59401, 75475, 95494, 119545, 149035, 184118, 226562, 276620, 336470, 406490, 489344, 585572, 698397, 828549, 979896

Offset: 0

N. J. A. Sloane

Also the number of ways in which to label the vertices of the cube (or faces of the octahedron) with nonnegative integers summing to n, where labelings that differ only by rotation or reflection are considered the same. - Isabel C. Lugo (izzycat(AT)gmail.com), Aug 26 2004

			Representatives of the seven classes of 3 X 3 binary matrices are:
[ 1 1 1 ] [ 1 1 0 ] [ 1 0 1 ] [ 1 0 1 ] [ 0 1 1 ] [ 0 1 1 ] [ 0 1 1 ]
[ 1 1 1 ] [ 1 1 1 ] [ 1 1 0 ] [ 1 1 0 ] [ 1 0 1 ] [ 1 0 0 ] [ 1 0 0 ]
[ 1 1 1 ] [ 1 1 1 ] [ 1 1 1 ] [ 1 1 0 ] [ 1 1 0 ] [ 1 1 1 ] [ 1 0 0 ].

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1052.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,2,-5,7,-4,2,2,-8,10,-8,2,2,-4,7,-5,2,-1,-2,3,-1).
Index entries for sequences related to binary matrices
Index entries for two-way infinite sequences

Column k=3 of A363349.

Cf. A000601, A005232, A006382, A006380, A002727, A006148.

Vec((1/(1 - x^1)^8 + 13/(1 - x^2)^4 + 6/(1 - x^1)^4/(1 - x^2)^2 + 12/(1 - x^4)^2 + 8/(1 - x^1)^2/(1 - x^3)^2 + 8/(1 - x^2)^1/(1 - x^6)^1)/48 + O(x^41)) \\ Andrew Howroyd, May 30 2023

G.f.: (1/(1 - x^1)^8 + 13/(1 - x^2)^4 + 6/(1 - x^1)^4/(1 - x^2)^2 + 12/(1 - x^4)^2 + 8/(1 - x^1)^2/(1 - x^3)^2 + 8/(1 - x^2)^1/(1 - x^6)^1)/48.

G.f.: (x^14 - 2*x^13 + 3*x^12 - 2*x^11 + 5*x^10 - 4*x^9 + 7*x^8 - 4*x^7 + 7*x^6 - 4*x^5 + 5*x^4 - 2*x^3 + 3*x^2 - 2*x + 1)/(x^6 - 1)/(x^2 + 1)^2/(x^2 + x + 1)/(x + 1)^3/(x - 1)^7.

Entry revised by Vladeta Jovovic, Aug 05 2000

Definition corrected by Max Alekseyev, Feb 05 2010

A363349 Array read by antidiagonals: T(n,k) is the number of equivalence classes of n X k binary matrices under permutation of rows and columns and complementation of columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 4, 3, 1, 1, 1, 5, 7, 8, 3, 1, 1, 1, 6, 11, 19, 10, 4, 1, 1, 1, 7, 16, 41, 32, 16, 4, 1, 1, 1, 8, 23, 81, 101, 68, 20, 5, 1, 1, 1, 9, 31, 153, 299, 301, 114, 29, 5, 1, 1, 1, 10, 41, 273, 849, 1358, 757, 210, 35, 6, 1

Offset: 0

Andrew Howroyd, May 28 2023

T(n,k) is also the number of equivalence classes of n X k binary matrices with an even number of 1's in each column under permutation of rows and columns.

			Array begins:
======================================================
n/k| 0 1  2   3    4     5      6       7        8 ...
---+--------------------------------------------------
0  | 1 1  1   1    1     1      1       1        1 ...
1  | 1 1  1   1    1     1      1       1        1 ...
2  | 1 2  3   4    5     6      7       8        9 ...
3  | 1 2  4   7   11    16     23      31       41 ...
4  | 1 3  8  19   41    81    153     273      468 ...
5  | 1 3 10  32  101   299    849    2290     5901 ...
6  | 1 4 16  68  301  1358   6128   27008   114763 ...
7  | 1 4 20 114  757  5567  43534  343656  2645494 ...
8  | 1 5 29 210 1981 23350 319119 4633380 67013431 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.
M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)

A259344 is the same array without the first row and column read by upward antidiagonals.
Columns k=0..6 are A000012, A004526(n+2), A005232, A006381, A006382, A056204, A056205.
Rows n=2..4 are A000027(n+1), A000601, A006380.
Main diagonal is A006383.
Cf. A028657, A241956, A362905.

\\ Compare A028657.
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t)={sum(j=1, #q, gcd(t, q[j]))}
T(n, k)={if(n==0, 1, my(s=0); forpart(q=n, my(e=1<

A006383 Number of equivalence classes of n X n binary matrices when one can permute rows, permute columns and complement columns.

Original entry on oeis.org

1, 1, 3, 7, 41, 299, 6128, 343656, 67013431, 45770163273, 108577103160005, 886929528971819040, 24943191706060101926577, 2425246700258693990625775794, 820270898724825121532156178527106

Offset: 0

N. J. A. Sloane

			a(2) = 3:
00 10 11
00 00 00

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 0..35 from Sean A. Irvine)
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.
M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)
Index entries for sequences related to binary matrices

Main diagonal of A363349.

Cf. A002724, A005232, A006381, A006382, A056204, A056205.

Definition corrected by Brendan McKay, Jan 07 2007

Terms a(7) onward from Max Alekseyev, Feb 05 2010

A056204 Number of n X 5 binary matrices under row and column permutations and column complementations.

Original entry on oeis.org

1, 1, 6, 16, 81, 299, 1358, 5567, 23350, 91998, 351058, 1269907, 4394634, 14495236, 45779246, 138567568, 403282017, 1130773069, 3062535192, 8028046724, 20411824364, 50429813556, 121280243676, 284360432241, 650972702410

Offset: 0

Vladeta Jovovic, Aug 05 2000

nonn

M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, order 148.

Column k=5 of A363349.

Cf. A005232, A006380, A006381, A006382, A002727, A006148, A052264.

G.f.: 1/3840*(1/(1 - x^1)^32 + 231/(1 - x^2)^16 + 20/(1 - x^1)^16/(1 - x^2)^8 + 520/(1 - x^4)^8 + 60/(1 - x^1)^8/(1 - x^2)^12 + 80/(1 - x^1)^8/(1 - x^3)^8 + 720/(1 - x^2)^4/(1 - x^6)^4 + 160/(1 - x^1)^4/(1 - x^2)^2/(1 - x^3)^4/(1 - x^6)^2 + 320/(1 - x^4)^2/(1 - x^12)^2 + 240/(1 - x^1)^4/(1 - x^2)^2/(1 - x^4)^6 + 480/(1 - x^8)^4 + 240/(1 - x^2)^4/(1 - x^4)^6 + 384/(1 - x^1)^2/(1 - x^5)^6 + 384/(1 - x^2)^1/(1 - x^10)^3).

A056205 Number of n X 6 binary matrices under row and column permutations and column complementations.

Original entry on oeis.org

1, 1, 7, 23, 153, 849, 6128, 43534, 319119, 2255466, 15307395, 98349144, 597543497, 3430839916, 18653684881, 96273409815, 473010823993, 2218614773950, 9961651259869, 42927432229913, 177963663264430

Offset: 0

Vladeta Jovovic, Aug 05 2000

nonn

M. A. Harrison, On the number of classes of binary matrices, IEEE Trans.Computers, 22 (1973), 1048-1051.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, order 312.

Column k=6 of A363349.

Cf. A005232, A006380, A006381, A006382, A002727, A006148, A052264.

G.f.: 1/46080*(1/(1 - x^1)^64 + 1053/(1 - x^2)^32 + 30/(1 - x^1)^32/(1 - x^2)^16 + 4920/(1 - x^4)^16 + 180/(1 - x^1)^16/(1 - x^2)^24 + 120/(1 - x^1)^8/(1 - x^2)^28 + 160/(1 - x^1)^16/(1 - x^3)^16 + 5280/(1 - x^2)^8/(1 - x^6)^8 + 960/(1 - x^1)^8/(1 - x^2)^4/(1 - x^3)^8/(1 - x^6)^4 + 3840/(1 - x^4)^4/(1 - x^12)^4 + 640/(1 - x^1)^4/(1 - x^3)^20 + 1920/(1 - x^2)^2/(1 - x^6)^10 + 720/(1 - x^1)^8/(1 - x^2)^4/(1 - x^4)^12 + 5760/(1 - x^8)^8 + 2160/(1 - x^2)^8/(1 - x^4)^12 + 1440/(1 - x^1)^4/(1 - x^2)^6/(1 - x^4)^12 + 2304/(1 - x^1)^4/(1 - x^5)^12 + 6912/(1 - x^2)^2/(1 - x^10)^6 + 3840/(1 - x^1)^2/(1 - x^2)^1/(1 - x^3)^2/(1 - x^6)^9 + 3840/(1 - x^4)^1/(1 - x^12)^5).

A363350 Number of n element multisets of length 4 vectors over GF(2) that sum to zero.

Original entry on oeis.org

1, 1, 16, 51, 276, 969, 3504, 10659, 30954, 81719, 205040, 482885, 1088100, 2340135, 4850640, 9694845, 18789795, 35357670, 64833120, 115997970, 203014680, 347993910, 585292320, 966955410, 1571349780, 2514084066, 3964589856, 6167026726, 9470900056, 14369476066, 21554373984

Offset: 0

Andrew Howroyd, May 30 2023

a(n) is the number of n X 4 binary matrices under row permutations and column complementations.

See A362905 for other interpretations.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8, -20, -8, 126, -168, -196, 680, -239, -1072, 1240, 560, -1820, 560, 1240, -1072, -239, 680, -196, -168, 126, -8, -20, 8, -1).

Column k=4 of A362905.

Cf. A006382.

A363350[n_]:=(Binomial[n+15,15]+If[EvenQ[n],15Binomial[n/2+7,7],0])/16;Array[A363350,50,0] (* Paolo Xausa, Nov 18 2023 *)

a(n) = (binomial(n+15,15) + if(n%2==0, 15*binomial(n/2+7, 7)))/16

G.f.: (1 - 7*x + 28*x^2 - 49*x^3 + 70*x^4 - 49*x^5 + 28*x^6 - 7*x^7 + x^8)/((1 - x)^16*(1 + x)^8).
a(n) = binomial(n+15, 15)/16 for odd n;
a(n) = (binomial(n+15, 15) + 15*binomial(n/2+7, 7))/16 for even n.

A259344 Array read by antidiagonals: number of inequivalent m X n (0,1)-matrices under permutation of rows and permutation and/or complementation of columns.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 4, 4, 3, 1, 5, 7, 8, 3, 1, 6, 11, 19, 10, 4, 1, 7, 16, 41, 32, 16, 4, 1, 8, 23, 81, 101, 68, 20, 5

Offset: 1

N. J. A. Sloane, Jun 27 2015

			The first few antidiagonals are:
1,
1,2,
1,3,2,
1,4,4,3,
1,5,7,8,3,
1,6,11,19,10,4,
1,7,16,41,32,16,4,
1,8,23,81,101,68,20,5,
...

M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051. See Table II.
M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)

For some rows, columns, diagonals see A006380, A006281, A006382, A006383.

The second row of the array starts 2,4,7,11,16,23, which does not identify it uniquely.

cp's OEIS Frontend

A005232 Expansion of (1-x+x^2)/((1-x)^2*(1-x^2)*(1-x^4)).

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A006381 Number of n X 3 binary matrices under row and column permutations and column complementations.

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A363349 Array read by antidiagonals: T(n,k) is the number of equivalence classes of n X k binary matrices under permutation of rows and columns and complementation of columns.

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A006383 Number of equivalence classes of n X n binary matrices when one can permute rows, permute columns and complement columns.

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A056204 Number of n X 5 binary matrices under row and column permutations and column complementations.

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A056205 Number of n X 6 binary matrices under row and column permutations and column complementations.

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A363350 Number of n element multisets of length 4 vectors over GF(2) that sum to zero.

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A259344 Array read by antidiagonals: number of inequivalent m X n (0,1)-matrices under permutation of rows and permutation and/or complementation of columns.

A005232 Expansion of (1-x+x^2)/((1-x)^2(1-x^2)(1-x^4)).