A039946 -id:A039946 - cp's OEIS frontend

A027633 Molien series for full 8 X 8 Siegel modular group H_3 of order 371589120.

1, 0, 1, 1, 2, 2, 5, 4, 9, 10, 16, 19, 31, 34, 53, 64, 89, 109, 152, 179, 245, 296, 384, 467, 601, 716, 911, 1090, 1351, 1614, 1986, 2342, 2856, 3364, 4037, 4742, 5653, 6578, 7791, 9036, 10592, 12243, 14268, 16380, 18990, 21724, 24999

Offset: 0

N. J. A. Sloane

			1 + x^4 + x^6 + 2*x^8 + 2*x^10 + 5*x^12 + 4*x^14 + 9*x^16 + 10*x^18 + 16*x^20 + ...

Jean-François Alcover, Table of n, a(n) for n = 0..999
B. Runge, On Siegel modular forms II, Nagoya Math. J., 138 (1995), 179-197.
Index entries for Molien series
Index entries for sequences related to modular groups
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 1, -1, 2, -1, -1, 1, -2, 0, 0, -2, 2, 0, -1, 3, -1, 1, 2, -3, 2, 0, -3, 3, -3, 0, 3, -4, 3, 0, -3, 3, -3, 0, 2, -3, 2, 1, -1, 3, -1, 0, 2, -2, 0, 0, -2, 1, -1, -1, 2, -1, 1, 0, 0, 1, -1).

Cf. A027672, A027638. Bisection gives A039946.

R. = PowerSeriesRing(ZZ,40);
g = 1 + x^4 + x^10 + 3*x^16 - x^18 + 3*x^20 + 2*x^22 + 2*x^24 + 3*x^26 + 4*x^28 + 2*x^30 + 7*x^32 + 3*x^34 + 7*x^36 + 5*x^38 + 9*x^40 + 6*x^42 + 10*x^44 + 8*x^46 + 9*x^50 + 7*x^54 - x^2 + 12*x^52 + 10*x^48 + 7*x^56;
f = g + x^112*g(1/x);
h = f(x)*(1 + x^2)/((1 - x^4)*(1 - x^8)*(1 - x^12)^2*(1 - x^14)*(1 - x^18)*(1 - x^20)*(1 - x^30));
[h.list()[2*i] for i in range(20)] # Andy Huchala, Mar 02 2022

Reference gives explicit formula for Molien series.
Molien series is f(x)*(1 + x^2)/((1 - x^4)*(1 - x^8)*(1 - x^12)^2*(1 - x^14)*(1 - x^18)*(1 - x^20)*(1 - x^30)),
where f(x) = g(x) + x^112*g(1/x), g(x) = 1 + x^4 + x^10 + 3*x^16 - x^18 + 3*x^20 + 2*x^22 + 2*x^24 + 3*x^26 + 4*x^28 + 2*x^30 + 7*x^32 + 3*x^34 + 7*x^36 + 5*x^38 + 9*x^40 + 6*x^42 + 10*x^44 + 8*x^46 + 9*x^50 + 7*x^54 - x^2 + 12*x^52 + 10*x^48 + 7*x^56.

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

A043330 Expansion of Molien series for relative invariants of 8-dimensional complex Clifford group.

Original entry on oeis.org

1, 2, 4, 10, 19, 34, 64, 109, 179, 296, 467, 716, 1090, 1614, 2342, 3364, 4742, 6578, 9036, 12243, 16380, 21724, 28501, 37017, 47713, 60968, 77264, 97271, 121588, 150946, 186326, 228627, 278912, 338576, 408923, 491451, 588089

Offset: 0

E. M. Rains

nonn

			t^12+2*t^20+4*t^28+10*t^36+19*t^44+34*t^52+64*t^60+109*t^68+179*t^76+...

Index entries for Molien series

Cf. A039946.

cp's OEIS Frontend

A027633 Molien series for full 8 X 8 Siegel modular group H_3 of order 371589120.

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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.

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A043330 Expansion of Molien series for relative invariants of 8-dimensional complex Clifford group.

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