A027672 -id:A027672 - cp's OEIS frontend

A003956 Order of complex Clifford group of degree 2^n arising in quantum coding theory.

8, 192, 92160, 743178240, 97029351014400, 203286581427673497600, 6819500449352277792129024000, 3660967964237442812098963052691456000, 31446995505814020383166371418359014222725120000

Offset: 0

N. J. A. Sloane and Peter Shor

T. D. Noe, Table of n, a(n) for n = 0..20
Simon Burton, Elijah Durso-Sabina, and Natalie C. Brown, Genons, Double Covers and Fault-tolerant Clifford Gates, arXiv:2406.09951 [quant-ph], 2024. See p. 18.
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. Inform. Theory, 44 (1998), 1369-1387.
G. Nebe, E. M. Rains, and N. J. A. Sloane, The invariants of the Clifford groups, arXiv:math/0001038 [math.CO], 2000; Des. Codes Crypt. 24 (2001), 99-121.
G. Nebe, E. M. Rains, and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Edwin Pednault, An alternative approach to optimal wire cutting without ancilla qubits, arXiv:2303.08287 [quant-ph], 2023.
Tefjol Pllaha, Olav Tirkkonen, and Robert Calderbank, Binary Subspace Chirps, arXiv:2102.12384 [cs.IT], 2021.
Bernhard Runge, Codes and Siegel modular forms, Discrete Math. 148 (1996), 175-204.
Peter Selinger, Generators and relations for n-qubit Clifford operators, arXiv:1310.6813 [quant-ph], 2013; Log. Methods Comput. Sci. 11 (2:10) (2015), 1-17, doi:10.2168/LMCS-11(2:10)2015.
Index entries for sequences related to groups.

Cf. A001309, A014116, A014115, A027672, A100221.

Equals twice A027638.

List([0..10], n-> 2^((n+1)^2 +2)*Product([1..n], j-> 4^j -1) ); # G. C. Greubel, Sep 24 2019

[n eq 0 select 8 else 2^((n+1)^2+2)*(&*[4^j-1: j in [1..n]]): n in [0..10]]; // G. C. Greubel, Sep 24 2019

a(n):= 2^(n^2+2*n+3)*mul(4^j-1, j=1..n); seq(a(n), n=0..10); # modified by G. C. Greubel, Sep 24 2019

Table[2^(n^2+2n+3) Product[4^j-1,{j,n}],{n,0,10}] (* Harvey P. Dale, Nov 03 2017 *)

vector(11, n, 2^(n^2 +2)*prod(j=1,n-1, 4^j-1) ) \\ G. C. Greubel, Sep 24 2019

from math import prod
def A003956(n): return prod((1<Chai Wah Wu, Jun 20 2022

[2^((n+1)^2 +2)*product(4^j -1 for j in (1..n)) for n in (0..10)] # G. C. Greubel, Sep 24 2019

From Amiram Eldar, Jul 06 2025: (Start)
a(n) = 2^(n^2+2*n+3) * Product_{k=1..n} (4^k-1).
a(n) ~ c * 2^(2*n^2+3*n+3), where c = A100221. (End)

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

Original entry on oeis.org

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.

A027638 Order of 2^n X 2^n unitary group H_n acting on Siegel modular forms.

Original entry on oeis.org

4, 96, 46080, 371589120, 48514675507200, 101643290713836748800, 3409750224676138896064512000, 1830483982118721406049481526345728000, 15723497752907010191583185709179507111362560000

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..39
Simon Burton, Elijah Durso-Sabina, and Natalie C. Brown, Genons, Double Covers and Fault-tolerant Clifford Gates, arXiv:2406.09951 [quant-ph], 2024. See p. 18.
Bernhard Runge, On Siegel modular forms. Part I, J. Reine Angew. Math., 436 (1993), 57-85.
Bernhard Runge, Codes and Siegel modular forms, Discrete Math. 148 (1996), 175-204.
Index entries for sequences related to modular groups.

Cf. A003956, A027672, A027639, A100221.

A027638:= func< n | n eq 0 select 4 else 2^(n^2+2*n+2)*(&*[4^j-1: j in [1..n]]) >;
[A027638(n): n in [0..15]]; // G. C. Greubel, Aug 04 2022

seq( 2^(n^2+2*n+2)*product(4^i -1, i=1..n), n=0..12);

Table[2^(n^2+2n+2) Product[4^k-1,{k,n}],{n,0,10}] (* Harvey P. Dale, May 21 2018 *)

a(n) = my(ret=1); for(i=1,n, ret = ret<<(2*i)-ret); ret << (n^2+2*n+2); \\ Kevin Ryde, Aug 13 2022

from sage.combinat.q_analogues import q_pochhammer
def A027638(n): return (-1)^n*2^(n^2 + 2*n + 2)*q_pochhammer(n, 4, 4)
[A027638(n) for n in (0..15)] # G. C. Greubel, Aug 04 2022

a(n) = A003956(n)/2.
a(n) = 2^(n^2 + 2*n + 2) * Product_{j=1..n} (4^j - 1).
a(n) ~ c * 2^(2*n^2+3*n+2), where c = A100221. - Amiram Eldar, Jul 06 2025

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

A144060 Expansion of Molien series for the ring of genus 5 code polynomials for Type II codes.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 8, 8, 34, 60, 203, 553, 2063, 7359, 30811, 127416, 541644, 2235677, 8966371, 34413747, 126465849, 443858877, 1490702752, 4796609651, 14821521743, 44071296447, 126388352322, 350298687803, 940211047828, 2448320130626, 6196158876181

Offset: 0

N. J. A. Sloane, Dec 22 2008, following a suggestion from G. Nebe

nonn

			1 + x^8 + 2*x^16 + 2*x^20 + 8*x^24 + 8*x^28 + 34*x^32 + 60*x^36 + 203*x^40 + 553*x^44 + 2063*x^48 + 7359*x^52 + 30811*x^56 + 127416*x^60 + 541644*x^64 + 2235677*x^68 + 8966371*x^72 + 34413747*x^76 + 126465849*x^80 + 443858877*x^84 + 1490702752*x^88 + 4796609651*x^92 + 14821521743*x^96 + .....

Ray Chandler, Table of n, a(n) for n = 0..2000
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.
M. Oura, The dimension formula of the invariant ring of H5 (web archive).
Index entries for linear recurrences with constant coefficients, order 532.

Cf. A027672, A051354.

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

Oura (see link) gives the Molien series explicitly.

More terms from Ray Chandler, Mar 23 2017

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A003956 Order of complex Clifford group of degree 2^n arising in quantum coding theory.

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

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A027638 Order of 2^n X 2^n unitary group H_n acting on Siegel modular forms.

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

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A144060 Expansion of Molien series for the ring of genus 5 code polynomials for Type II codes.

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