cp's OEIS Frontend

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

%I A027638 #31 Jul 07 2025 03:41:59
%S A027638 4,96,46080,371589120,48514675507200,101643290713836748800,
%T A027638 3409750224676138896064512000,1830483982118721406049481526345728000,
%U A027638 15723497752907010191583185709179507111362560000
%N A027638 Order of 2^n X 2^n unitary group H_n acting on Siegel modular forms.
%H A027638 G. C. Greubel, <a href="/A027638/b027638.txt">Table of n, a(n) for n = 0..39</a>
%H A027638 Simon Burton, Elijah Durso-Sabina, and Natalie C. Brown, <a href="https://arxiv.org/abs/2406.09951">Genons, Double Covers and Fault-tolerant Clifford Gates</a>, arXiv:2406.09951 [quant-ph], 2024. See p. 18.
%H A027638 Bernhard Runge, <a href="https://eudml.org/doc/153497">On Siegel modular forms. Part I</a>, J. Reine Angew. Math., 436 (1993), 57-85.
%H A027638 Bernhard Runge, <a href="https://doi.org/10.1016/0012-365X(94)00271-J">Codes and Siegel modular forms</a>, Discrete Math. 148 (1996), 175-204.
%H A027638 <a href="/index/Gre#groups_modular">Index entries for sequences related to modular groups</a>.
%F A027638 a(n) = A003956(n)/2.
%F A027638 a(n) = 2^(n^2 + 2*n + 2) * Product_{j=1..n} (4^j - 1).
%F A027638 a(n) ~ c * 2^(2*n^2+3*n+2), where c = A100221. - _Amiram Eldar_, Jul 06 2025
%p A027638 seq( 2^(n^2+2*n+2)*product(4^i -1, i=1..n), n=0..12);
%t A027638 Table[2^(n^2+2n+2) Product[4^k-1,{k,n}],{n,0,10}] (* _Harvey P. Dale_, May 21 2018 *)
%o A027638 (Magma)
%o A027638 A027638:= func< n | n eq 0 select 4 else 2^(n^2+2*n+2)*(&*[4^j-1: j in [1..n]]) >;
%o A027638 [A027638(n): n in [0..15]]; // _G. C. Greubel_, Aug 04 2022
%o A027638 (SageMath)
%o A027638 from sage.combinat.q_analogues import q_pochhammer
%o A027638 def A027638(n): return (-1)^n*2^(n^2 + 2*n + 2)*q_pochhammer(n, 4, 4)
%o A027638 [A027638(n) for n in (0..15)] # _G. C. Greubel_, Aug 04 2022
%o A027638 (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
%Y A027638 Cf. A003956, A027672, A027639, A100221.
%K A027638 nonn,easy,nice
%O A027638 0,1
%A A027638 _N. J. A. Sloane_