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A027639 Order of unitary 2^n X 2^n group H_{n,4} acting on Siegel modular forms.

%I A027639 #33 Jul 10 2025 04:17:18
%S A027639 4,32,3072,2752512,21139292160,1342091380654080,692647993190048071680,
%T A027639 2882479558988139892026900480,96342151992701835341576224427212800,
%U A027639 25811138467998276182105365247324712232550400
%N A027639 Order of unitary 2^n X 2^n group H_{n,4} acting on Siegel modular forms.
%H A027639 G. C. Greubel, <a href="/A027639/b027639.txt">Table of n, a(n) for n = 0..45</a>
%H A027639 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 A027639 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 A027639 <a href="/index/Gre#groups_modular">Index entries for sequences related to modular groups</a>.
%F A027639 a(n) = 2^(n^2 + 2*n + 2) * Product_{j=1..n} (2^j - 1).
%F A027639 a(n) = (-1)^n * 2^(n^2 + 2*n + 2) * (2, 2)_{n}, where (q, q)_{n} is the q-Pochhammer symbol. - _G. C. Greubel_, Aug 04 2022
%F A027639 a(n) ~ c * 2^((3*n^2+5*n+4)/2), where c = A048651. - _Amiram Eldar_, Jul 09 2025
%p A027639 seq(2^(n^2+2*n+2)*mul(2^i-1, i=1..n), n=0..10);
%t A027639 a[n_]:= (-1)^n*2^(n^2 +2*n+2)*QPochhammer[2,2,n];
%t A027639 Table[a[n], {n, 0, 15}] (* _G. C. Greubel_, Aug 04 2022 *)
%o A027639 (Magma)
%o A027639 A027639:= func< n | n eq 0 select 4 else 2^(n^2+2*n+2)*(&*[2^j-1: j in [1..n]]) >;
%o A027639 [A027639(n): n in [0..15]]; // _G. C. Greubel_, Aug 04 2022
%o A027639 (SageMath)
%o A027639 from sage.combinat.q_analogues import q_pochhammer
%o A027639 def A027639(n): return (-1)^n*2^(n^2 + 2*n + 2)*q_pochhammer(n, 2, 2)
%o A027639 [A027639(n) for n in (0..15)] # _G. C. Greubel_, Aug 04 2022
%o A027639 (PARI) a(n) = my(ret=1); for(i=1,n, ret = ret<<i-ret); ret << (n^2 + 2*n + 2); \\ _Kevin Ryde_, Aug 13 2022
%Y A027639 Cf. A005329, A048651.
%K A027639 nonn,nice,easy
%O A027639 0,1
%A A027639 _N. J. A. Sloane_