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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Links

• 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.

Crossrefs

Cf. A003956, A027672, A027639, A100221.

Programs

Magma

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

Maple

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

Mathematica

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

PARI

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

SageMath

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

Formula

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