A003956 -id:A003956 - cp's OEIS frontend

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

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

A353196 Number of stabilizer states on n qubits.

Original entry on oeis.org

6, 60, 1080, 36720, 2423520, 315057600, 81284860800, 41780418451200, 42866709330931200, 87876754128408960000, 360118938418219918080000, 2950814581398894008747520000, 48352047730802277227336862720000, 1584496604138390624739828991334400000

Offset: 1

James Rayman, Apr 29 2022

A stabilizer state is a quantum state on n qubits prepared by applying a series of Hadamard, CNOT, and S gates to the all-zero state. There are only a finite number of such states for any n.

			For n = 1, the a(1) = 6 states are |0>, |1>, |+>, |->, |i>, and |-i>.

D. Gross, Hudson's Theorem for finite-dimensional quantum systems, arXiv:quant-ph/0602001, 2006-2007.

Cf. A000079, A003956, A028362.

Table[2^n * QPochhammer[-2, 2, n], {n, 13}] (* Amiram Eldar, Aug 17 2025 *)

def a(n):
    ans = 2 ** n
    for i in range(1, n+1):
        ans *= 2 ** i + 1
    return ans

from math import prod
def A353196(n): return prod((1<Chai Wah Wu, Jun 20 2022

a(n) = 2^n*Product_{i=1..n} (2^i+1).
a(n) = A000079(n)*A028362(n+1).
a(n) ~ c * 2^(n*(n+3)/2) where c = Product_{k>=1} (1 + 1/2^k) = A079555. - Amiram Eldar, Aug 17 2025

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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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A353196 Number of stabilizer states on n qubits.

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