A008718 -id:A008718 - cp's OEIS frontend

A008621 Expansion of 1/((1-x)*(1-x^4)).

1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21

Offset: 0

N. J. A. Sloane

Arises from Gleason's theorem on self-dual codes: 1/((1-x^2)*(1-x^8)) is the Molien series for the real 2-dimensional Clifford group (a dihedral group of order 16) of genus 1.
Thickness of the hypercube graph Q_n. - Eric W. Weisstein, Sep 09 2008
Count of odd numbers between consecutive quarter-squares, A002620. Oppermann's conjecture states that for each count there will be at least one prime. - Fred Daniel Kline, Sep 10 2011
Number of partitions into parts 1 and 4. - Joerg Arndt, Jun 01 2013
a(n-1) is the minimum independence number over all planar graphs with n vertices. The bound follows from the Four Color Theorem. It is attained by a union of 4-cliques. Other extremal graphs are examined in the Bickle link. - Allan Bickle, Feb 04 2022

D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
F. J. MacWilliams and N. J. A. Sloane, Theory of Error-Correcting Codes, 1977, Chapter 19, Problem 3, p. 602.

T. D. Noe, Table of n, a(n) for n = 0..1000
Allan Bickle, Independence Number of Maximal Planar Graphs, Congr. Num. 234 (2019) 61-68.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 211
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Eric Weisstein's World of Mathematics, Graph Thickness
Wikipedia, Oppermann's conjecture
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A002265 (equals this - 1).

Cf. A008718, A024186, A110160, A110868, A110869, A110876, A110880, A008620.

Table[Floor[n/4]+1, {n, 0, 80}] (* Stefan Steinerberger, Apr 03 2006 *)
CoefficientList[Series[1/((1-x)(1-x^4)),{x,0,80}],x] (* Harvey P. Dale, Feb 19 2012 *)
Flatten[ Table[ PadRight[{},4,n],{n,19}]] (* Harvey P. Dale, Feb 19 2012 *)

a(n)=n\4+1 \\ Charles R Greathouse IV, Feb 06 2017

[n//4+1 for n in range(85)] # Gennady Eremin, Mar 01 2022

a(n) = floor(n/4) + 1.
a(n) = A010766(n+4, 4).
Also, a(n) = ceiling((n+1)/4), n >= 0. - Mohammad K. Azarian, May 22 2007
a(n) = Sum_{i=0..n} A121262(i) = n/4 + 5/8 + (-1)^n/8 + A057077(n)/4. - R. J. Mathar, Mar 14 2011
a(x,y) := floor(x/2) + floor(y/2) - x where x = A002620(n) and y = A002620(n+1), n > 2. - Fred Daniel Kline, Sep 10 2011
a(n) = a(n-1) + a(n-4) - a(n-5); a(0)=1, a(1)=1, a(2)=1, a(3)=1, a(4)=2. - Harvey P. Dale, Feb 19 2012
From R. J. Mathar, Jun 04 2021: (Start)
G.f.: 1 / ( (1+x)*(1+x^2)*(x-1)^2 ).
a(n) + a(n-1) = A004524(n+3).
a(n) + a(n-2) = A008619(n). (End)
a(n) = A002265(n) + 1. - M. F. Hasler, Oct 17 2022

More terms from Stefan Steinerberger, Apr 03 2006

A024186 Expansion of Molien series for 8-dimensional real Clifford group 2^{1+6}.Alt_8.2 of genus 3 and order 5160960.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 12, 18, 22, 29, 35, 48, 57, 74, 91, 116, 140, 177, 211, 265, 319, 389, 462, 566, 667, 804, 949, 1131, 1324, 1573, 1827, 2153, 2502, 2917, 3364, 3916, 4491, 5187, 5937, 6813, 7760, 8879, 10058, 11448, 12950, 14658, 16500, 18632, 20894

Offset: 0

N. J. A. Sloane, G. Nebe (nebe(AT)math.rwth-aachen.de)

Jean-François Alcover, Table of n, a(n) for n = 0..999
A. R. Calderbank, R. H. Hardin, E. M. Rains, P. W. Shor, and N. J. A. Sloane, A Group-Theoretic Framework for the Construction of Packings in Grassmannian Spaces, J. Algebraic Combinatorics, 9 (1999), 129-140; arXiv:math/0208002 [math.CO], 2002.
A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, Quantum Error Correction Via Codes Over GF(4), arXiv:quant-ph/9608006, 1996-1997; IEEE Trans. Information Theory, 44 (1998), 1369-1387.
J. H. Conway, R. H. Hardin, and N. J. A. Sloane, Packing Lines, Planes, etc.: Packings in Grassmannian Space, Experimental Math. 5 (1996), 139-159; arXiv:math/0208004 [math.CO], 2002.
G. Nebe, E. M. Rains, and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, -1, 0, 0, 1, 0, -1, -1, 0, 3, -2, 0, -2, 2, 0, 0, -2, 1, 1, 1, -1, -1, 2, 0, -1, -1, 1, 2, 0, -3, 0, 2, 2, -3, -2, 0, 4, 0, -2, -3, 2, 2, 0, -3, 0, 2, 1, -1, -1, 0, 2, -1, -1, 1, 1, 1, -2, 0, 0, 2, -2, 0, -2, 3, 0, -1, -1, 0, 1, 0, 0, -1, 1, 0, 1, -1).

Cf. A008621, A008718.

// Commands to generate the group.
F := QuadraticField(2); M := GeneralLinearGroup(8, F); t := 1/(2*s);
B := M! [ -t, -t, -t, -t, -t, -t, -t, -t,
-t, t, -t, -t, t, -t, t, t,
-t, -t, -t, t, -t, t, t, t,
-t, -t, t, -t, t, t, t, -t,
-t, t, -t, t, t, t, -t, -t,
-t, -t, t, t, t, -t, -t, t,
-t, t, t, t, -t, -t, t, -t,
-t, t, t, -t, -t, t, -t, t ];
S := M! [ -1, 0, 0, 0, 0, 0, 0, 0,
0, -1, 0, 0, 0, 0, 0, 0,
0, 0, 1, 0, 0, 0, 0, 0,
0, 0, 0, 1, 0, 0, 0, 0,
0, 0, 0, 0, 1, 0, 0, 0,
0, 0, 0, 0, 0, 1, 0, 0,
0, 0, 0, 0, 0, 0, 1, 0,
0, 0, 0, 0, 0, 0, 0, 1 ];
C := M! [ 1, 0, 0, 0, 0, 0, 0, 0,
0, 0, 1, 0, 0, 0, 0, 0,
0, 0, 0, 1, 0, 0, 0, 0,
0, 0, 0, 0, 1, 0, 0, 0,
0, 0, 0, 0, 0, 1, 0, 0,
0, 0, 0, 0, 0, 0, 1, 0,
0, 0, 0, 0, 0, 0, 0, 1,
0, 1, 0, 0, 0, 0, 0, 0 ];
G := sub< M | B, S, C >; Order(G);

Mathematica

ker = {1, 0, 1, -1, 0, 0, 1, 0, -1, -1, 0, 3, -2, 0, -2, 2, 0, 0, -2, 1, 1, 1, -1, -1, 2, 0, -1, -1, 1, 2, 0, -3, 0, 2, 2, -3, -2, 0, 4, 0, -2, -3, 2, 2, 0, -3, 0, 2, 1, -1, -1, 0, 2, -1, -1, 1, 1, 1, -2, 0, 0, 2, -2, 0, -2, 3, 0, -1, -1, 0, 1, 0, 0, -1, 1, 0, 1, -1}; init = {1, 1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 12, 18, 22, 29, 35, 48, 57, 74, 91, 116, 140, 177, 211, 265, 319, 389, 462, 566, 667, 804, 949, 1131, 1324, 1573, 1827, 2153, 2502, 2917, 3364, 3916, 4491, 5187, 5937, 6813, 7760, 8879, 10058, 11448, 12950, 14658, 16500, 18632, 20894, 23487, 26279, 29417, 32801, 36630, 40695, 45285, 50223, 55690, 61559, 68119, 75092, 82841, 91141, 100256, 110026, 120800, 132226, 144804, 158251, 172881, 188489, 205560, 223657}; LinearRecurrence[ker, init, 1000] (* Jean-François Alcover, Jan 05 2020 *)

Formula

Molien series = (t^148 - t^142 + t^140 + t^136 - t^134 + t^132 + 3*t^128 + 2*t^124 + 4*t^120 + 5*t^116 + 7*t^112 + t^110 + 7*t^108 + t^106 + 10*t^104 + 2*t^102 + 11*t^100 + 3*t^98 + 12*t^96 + 4*t^94 + 14*t^92 + 5*t^90 + 16*t^88 + 5*t^86 + 15*t^84 + 4*t^82 + 20*t^80 + 7*t^78 + 18*t^76 + 4*t^74 + 18*t^72 + 7*t^70 + 20*t^68 + 4*t^66 + 15*t^64 + 5*t^62 + 16*t^60 + 5*t^58 + 14*t^56 + 4*t^54 + 12*t^52 + 3*t^50 + 11*t^48 + 2*t^46 + 10*t^44 + t^42 + 7*t^40 + t^38 + 7*t^36 + 5*t^32 + 4*t^28 + 2*t^24 + 3*t^20 + t^16 - t^14 + t^12 + t^8 - t^6 + 1) /

(t^156 - t^154 - t^150 + t^148 - t^142 + t^138 + t^136 - 3*t^132 + 2*t^130 + 2*t^126 - 2*t^124 + 2*t^118 - t^116 - t^114 - t^112 + t^110 + t^108 - 2*t^106 + t^102 + t^100 - t^98 - 2*t^96 + 3*t^92 - 2*t^88 - 2*t^86 + 3*t^84 + 2*t^82 - 4*t^78 + 2*t^74 + 3*t^72 - 2*t^70 - 2*t^68 + 3*t^64 - 2*t^60 - t^58 + t^56 + t^54 - 2*t^50 + t^48 + t^46 - t^44 - t^42 - t^40 + 2*t^38 - 2*t^32 + 2*t^30 + 2*t^26 - 3*t^24 + t^20 + t^18 - t^14 + t^8 - t^6 - t^2 + 1).

Extensions

Rechecked Mar 30 2004. There were errors in the formula line, although not in the sequence itself.

A028288 Molien series for complex 4-dimensional Clifford group of order 92160 and genus 2. Also Molien series of ring of biweight enumerators of Type II self-dual binary codes.

Original entry on oeis.org

1, 1, 1, 3, 4, 5, 8, 10, 12, 17, 21, 24, 31, 37, 42, 52, 60, 67, 80, 91, 101, 117, 131, 144, 164, 182, 198, 222, 244, 264, 293, 319, 343, 377, 408, 437, 476, 512, 546, 591, 633, 672, 723, 771, 816, 874, 928, 979, 1044, 1105, 1163, 1235, 1303, 1368

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..1000
A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over GF(4), IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
W. Duke, On codes and Siegel modular forms, Int. Math. Res. Notes 1993, No. 5, Theorem 2.
W. C. Huffman, The biweight enumerator of self-orthogonal binary codes, Discr. Math. Vol. 26 1979, pp. 129-143.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,1,-2,1,-2,2,0,1,-1).

Cf. A008621, A008718, A024186, A039946, A051263.

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1+x^4)/((1-x)*(1-x^3)^2*(1-x^5)) )); // G. C. Greubel, Feb 01 2020

seq(coeff(series((1+x^4)/((1-x)*(1-x^3)^2*(1-x^5)), x, n+1), x, n), n = 0..60); # G. C. Greubel, Feb 01 2020

LinearRecurrence[{1,0,2,-2,1,-2,1,-2,2,0,1,-1}, {1,1,1,3,4,5,8,10,12,17,21,24}, 60] (* Jean-François Alcover, Jan 27 2015 *)
CoefficientList[Series[(1+x^4)/((1-x)(1-x^3)^2(1-x^5)),{x,0,60}],x] (* Harvey P. Dale, Jul 10 2019 *)

Vec((1+x^4)/((1-x)*(1-x^3)^2*(1-x^5)) + O('x^60)) \\ G. C. Greubel, Feb 01 2020

def A028288_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^4)/((1-x)*(1-x^3)^2*(1-x^5)) ).list()
A028288_list(60) # G. C. Greubel, Feb 01 2020

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

a(n) ~ 1/135*n^3. - Ralf Stephan, Apr 29 2014

A039946 Expansion of Molien series for 8-dimensional complex Clifford group of genus 3 and order 743178240.

Original entry on oeis.org

1, 1, 2, 5, 9, 16, 31, 53, 89, 152, 245, 384, 601, 911, 1351, 1986, 2856, 4037, 5653, 7791, 10592, 14268, 18990, 24999, 32643, 42218, 54112, 68869, 86971, 109014, 135812, 168101, 206769, 252990, 307849, 372616, 448934, 538348

Offset: 0

E. M. Rains

			G.f. = 1 + x^8 + 2*x^16 + 5*x^24 + 9*x^32 + 16*x^40 + 31*x^48 + ...

Jean-François Alcover, Table of n, a(n) for n = 0..499
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (1,1,1,-2,-1,0,1,-1,1,0,0,-1,1,2,1,-3,-2,0,2,1,-1,0,0,-1,1,2,0,-2,-3,1,2,1,-1,0,0,1,-1,1,0,-1,-2,1,1,1,-1).

Bisection of A027633. Cf. A008621, A008718, A024186, A008620, A028288, A043330, A051354.

f(x):= (1 +x^3 +3*x^4 +3*x^5 +6*x^6 +8*x^7 +12*x^8 +18*x^9 +25*x^10 +29*x^11 +40*x^12 +50*x^13 +58*x^14 +69*x^15 +80*x^16 +85*x^17 +96*x^18 +104*x^19 +107*x^20 +109*x^21 +112*x^22 +109*x^23+107*x^24 +104*x^25 +96*x^26 +85*x^27 +80*x^28 +69*x^29 +58*x^30 +50*x^31 +40*x^32 +29*x^33 +25*x^34 +18*x^35 +12*x^36 +8*x^37 +6*x^38 +3*x^39 +3*x^40 +x^41 +x^44) / ( (1-x)^2*(1-x^3)^4*(1-x^5)^2*(1 +x +2*x^3 +2*x^4 + x^5 +4*x^6 +2*x^7 +x^8 +5*x^9 +2*x^10 +2*x^11 +5*x^12 +x^13 +2*x^14 + 4*x^15 +x^16 +2*x^17 +2*x^18 +x^20 +x^21) ); seq(coeff(series(f(x), x, n+1), x, n), n = 0..40);

CoefficientList[Series[(1+x^3+3*x^4+3*x^5+6*x^6+8*x^7+12*x^8+18*x^9+25*x^10 + 29*x^11+40*x^12+50*x^13+58*x^14+69*x^15+80*x^16+85*x^17+96*x^18+104*x^19 + 107*x^20+109*x^21+112*x^22+109*x^23+107*x^24+104*x^25+96*x^26+85*x^27+80*x^28 +69*x^29+58*x^30+50*x^31+40*x^32+29*x^33+25*x^34+18*x^35+12*x^36 + 8*x^37 + 6*x^38+3*x^39+3*x^40+x^41+x^44)/((1-x)^2*(1-x^3)^4*(1-x^5)^2*(1+x+2*x^3+2*x^4 +x^5+4*x^6+2*x^7+x^8+5*x^9+2*x^10+2*x^11+5*x^12+x^13+2*x^14+4*x^15+x^16+2*x^17 +2*x^18+x^20+x^21)), {x,0,40}], x] (* G. C. Greubel, Feb 01 2020 *)
LinearRecurrence[{1,1,1,-2,-1,0,1,-1,1,0,0,-1,1,2,1,-3,-2,0,2,1,-1,0,0,-1,1,2,0,-2,-3,1,2,1,-1,0,0,1,-1,1,0,-1,-2,1,1,1,-1},{1,1,2,5,9,16,31,53,89,152,245,384,601,911,1351,1986,2856,4037,5653,7791,10592,14268,18990,24999,32643,42218,54112,68869,86971,109014,135812,168101,206769,252990,307849,372616,448934,538348,642630,764021,904658,1066943,1253876,1468340,1713529},40] (* Harvey P. Dale, Jul 04 2021 *)

G.f.: (1 +x^24 +3*x^32 +3*x^40 +6*x^48 +8*x^56 +12*x^64 +18*x^72 +25*x^80 +29*x^88 +40*x^96 +50*x^104 +58*x^112 +69*x^120 +80*x^128 +85*x^136 +96*x^144 +104*x^152 +107*x^160 +109*x^168 +112*x^176 +109*x^184 +107*x^192 +104*x^200 +96*x^208 +85*x^216 +80*x^224 +69*x^232 +58*x^240 +50*x^248 +40*x^256 +29*x^264 +25*x^272 +18*x^280 +12*x^288 +8*x^296 +6*x^304 +3*x^312 +3*x^320 +x^328 +x^352) / ( (1-x^8)^2*(1-x^24)^4*(1-x^40)^2*(1 +x^8 +2*x^24 +2*x^32 + x^40 +4*x^48 +2*x^56 +x^64 +5*x^72 +2*x^80 +2*x^88 +5*x^96 +x^104 +2*x^112 + 4*x^120 +x^128 +2*x^136 +2*x^144 +x^160 +x^168) ), nonzero terms.

G.f.: (1 +x^3 +3*x^4 +3*x^5 +6*x^6 +8*x^7 +12*x^8 +18*x^9 +25*x^10 +29*x^11 +40*x^12 +50*x^13 +58*x^14 +69*x^15 +80*x^16 +85*x^17 +96*x^18 +104*x^19 +107*x^20 +109*x^21 +112*x^22 +109*x^23+107*x^24 +104*x^25 +96*x^26 +85*x^27 +80*x^28 +69*x^29 +58*x^30 +50*x^31 +40*x^32 +29*x^33 +25*x^34 +18*x^35 +12*x^36 +8*x^37 +6*x^38 +3*x^39 +3*x^40 +x^41 +x^44) / ( (1-x)^2*(1-x^3)^4*(1-x^5)^2*(1 +x +2*x^3 +2*x^4 + x^5 +4*x^6 +2*x^7 +x^8 +5*x^9 +2*x^10 +2*x^11 +5*x^12 +x^13 +2*x^14 + 4*x^15 +x^16 +2*x^17 +2*x^18 +x^20 +x^21) ). - G. C. Greubel, Feb 01 2020

Typo in reduced g.f.s. corrected by Georg Fischer, Apr 18 2020

A051354 Expansion of Molien series for 16-dimensional complex Clifford group of genus 4 and order 97029351014400.

Original entry on oeis.org

1, 1, 2, 7, 19, 52, 172, 550, 1782, 5845, 18508, 56345, 164157, 454518, 1196924, 3003750, 7198311, 16523847, 36447873, 77478005, 159172517, 316874035, 612729396, 1153359711, 2117566545, 3798941401, 6670327291, 11479693332, 19390588953, 32185179449, 52553840336

Offset: 0

N. J. A. Sloane

Oura gives an explicit formula for the Molien series that produces A027672; the present sequence is the subsequence formed from the terms whose exponents are multiples of 8 (that is, every other term of A027672). In other words, the present Molien series is (f(x)+f(z*x))/2, where z = exp(2*Pi*I/8) and f(x) is the Molien series for the group H_4 given explicitly by Oura in Theorem 4.1.

			1 + t^8 + 2*t^16 + 7*t^24 + 19*t^32 + 52*t^40 + 172*t^48 + ...

Ray Chandler, Table of n, a(n) for n = 0..1000
Ray Chandler, Mathematica program
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
M. Oura, The dimension formula for the ring of code polynomials in genus 4, Osaka J. Math. 34 (1997), 53-72.
Index entries for linear recurrences with constant coefficients, signature (3, -2, 0, -3, 5, -2, -1, -1, 8, -7, -2, 2, 7, -7, -3, 1, 9, -11, 4, 3, 5, -9, -1, 0, 13, -15, 0, 4, 6, -9, 0, 8, 9, -18, -2, 12, -4, -4, -3, 6, 6, -8, -7, 18, -1, -6, -13, 13, 6, -14, -10, 30, -10, -10, -4, 22, -5, -6, -15, 28, -15, -6, -5, 22, -4, -10, -10, 30, -10, -14, 6, 13, -13, -6, -1, 18, -7, -8, 6, 6, -3, -4, -4, 12, -2, -18, 9, 8, 0, -9, 6, 4, 0, -15, 13, 0, -1, -9, 5, 3, 4, -11, 9, 1, -3, -7, 7, 2, -2, -7, 8, -1, -1, -2, 5, -3, 0, -2, 3, -1).
Index entries for Molien series

Cf. A027672, A003956, A008621, A008718, A024186, A008620, A028288, A043330.

Mathematica
```
(* See link for Mathematica program. *)
```

a(n) = A027672(2*n).

Edited by Georg Fischer, Jan 24 2021

A097913 G.f.: (1+x^18)/((1-x)(1-x^8)(1-x^12)*(1-x^24)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 6, 6, 6, 6, 9, 9, 10, 10, 11, 11, 12, 12, 15, 15, 16, 16, 19, 19, 20, 20, 23, 23, 26, 26, 29, 29, 30, 30, 36, 36, 39, 39, 42, 42, 45, 45, 51, 51, 54, 54, 60, 60, 63, 63, 69, 69, 75, 75, 81, 81, 84, 84, 94, 94, 100, 100, 106, 106

Offset: 0

N. J. A. Sloane, Sep 04 2004

nonn

Conjectured Poincaré series [or Poincare series] for genus 2 Siegel theta series of odd unimodular lattices.

G. C. Greubel, Table of n, a(n) for n = 0..1000
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1).

Cf. A008718.

CoefficientList[Series[(1 + x^18)/((1 - x)*(1 - x^8)*(1 - x^12)*(1 - x^24)), {x, 0, 50}], x] (* G. C. Greubel, Dec 20 2017 *)

x='x+O('x^30); Vec((1+x^18)/((1-x)*(1-x^8)*(1-x^12)*(1-x^24))) \\ G. C. Greubel, Dec 20 2017

A110868 Molien series for real 32-dimensional Clifford group of genus 5 and order 96253116206284800.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 7, 9, 16, 23, 46, 74

Offset: 0

G. Nebe, Sep 21 2005

nonn

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Index entries for Molien series

Cf. A008621, A008718, A024186, A110160, this sequence, A110869, A110876, A110880. See also A001309.

cp's OEIS Frontend

A008621 Expansion of 1/((1-x)*(1-x^4)).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A024186 Expansion of Molien series for 8-dimensional real Clifford group 2^{1+6}.Alt_8.2 of genus 3 and order 5160960.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A028288 Molien series for complex 4-dimensional Clifford group of order 92160 and genus 2. Also Molien series of ring of biweight enumerators of Type II self-dual binary codes.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A039946 Expansion of Molien series for 8-dimensional complex Clifford group of genus 3 and order 743178240.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A051354 Expansion of Molien series for 16-dimensional complex Clifford group of genus 4 and order 97029351014400.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A097913 G.f.: (1+x^18)/((1-x)*(1-x^8)*(1-x^12)*(1-x^24)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A110868 Molien series for real 32-dimensional Clifford group of genus 5 and order 96253116206284800.

Views

Author

Keywords

Links

Crossrefs

A097913 G.f.: (1+x^18)/((1-x)(1-x^8)(1-x^12)*(1-x^24)).