A027633 -id:A027633 - cp's OEIS frontend

A027672 Molien series for unitary 16-dimensional full Siegel modular group H_4 of order 48514675507200.

1, 0, 1, 1, 2, 3, 7, 7, 19, 27, 52, 87, 172, 279, 550, 960, 1782, 3183, 5845, 10288, 18508, 32284, 56345, 96473, 164157, 274194, 454518, 741321, 1196924, 1906123, 3003750, 4673470, 7198311, 10959836, 16523847, 24654860, 36447873, 53369530, 77478005, 111498073

Offset: 0

N. J. A. Sloane

			1+x^8+x^12+2*x^16+3*x^20+7*x^24+7*x^28+19*x^32+27*x^36+O(x^40).

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.
Bernhard Runge, On Siegel modular forms, part II, Nagoya Math. J. 138, 179-197 (1995).
Index entries for linear recurrences with constant coefficients, order 168.
Index entries for Molien series
Index entries for sequences related to modular groups

Cf. A027633, A027638, A051354.

Mathematica
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(* See link for Mathematica program. *)
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Oura gives an explicit formula for the Molien series.

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

A027634 Poincaré (or Molien) series for ring of Siegel modular forms of genus 3 (associated with full modular group Gamma_3).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 4, 3, 7, 8, 11, 15, 22, 24, 37, 45, 58, 75, 99, 115, 156, 187, 232, 288, 356, 420, 527, 623, 750, 898, 1075, 1252, 1505, 1750, 2051, 2400, 2797, 3214, 3754, 4294, 4939, 5665, 6477, 7344, 8398, 9481, 10731, 12121, 13653

Offset: 0

N. J. A. Sloane

			1+x^4+x^6+x^8+2*x^10+4*x^12+3*x^14+7*x^16+8*x^18+11*x^20+...

Andy Huchala, Table of n, a(n) for n = 0..20000
B. Runge, On Siegel modular forms II, Nagoya Math. J., 138 (1995), 179-197.
S. Tsuyumine, On Siegel modular forms of degree three, Amer. J. Math., 108 (1986), 755-862 and Addendum, Amer. J. Math., 108 (1986), 1001-1003.
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, 0, 0, 2, -1, -1, 1, 0, -1, -1, -1, 2, -1, -2, 2, 1, 0, 0, -1, 3, 0, -3, 2, 0, 0, 0, -2, 3, 0, -3, 1, 0, 0, -1, -2, 2, 1, -2, 1, 1, 1, 0, -1, 1, 1, -2, 0, 0, 0, 0, -1, 1).

Cf. A027633.

R. = PowerSeriesRing(ZZ,80);
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 = (1-x^8)*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(40)] # Andy Huchala, Mar 02 2022

(1-x^8) times Molien series in A027633. That is, same numerator, but denominator is (1-x^4)*(1-x^12)^2*(1-x^14)*(1-x^18)*(1-x^20)*(1-x^30).

cp's OEIS Frontend

A027672 Molien series for unitary 16-dimensional full Siegel modular group H_4 of order 48514675507200.

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

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A027634 Poincaré (or Molien) series for ring of Siegel modular forms of genus 3 (associated with full modular group Gamma_3).

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