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

4, 32, 3072, 2752512, 21139292160, 1342091380654080, 692647993190048071680, 2882479558988139892026900480, 96342151992701835341576224427212800, 25811138467998276182105365247324712232550400

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..45
• 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. A005329, A048651.

Programs

Magma

A027639:= func< n | n eq 0 select 4 else 2^(n^2+2*n+2)*(&*[2^j-1: j in [1..n]]) >;
[A027639(n): n in [0..15]]; // G. C. Greubel, Aug 04 2022

Maple

seq(2^(n^2+2*n+2)*mul(2^i-1, i=1..n), n=0..10);

Mathematica

a[n_]:= (-1)^n*2^(n^2 +2*n+2)*QPochhammer[2,2,n];
Table[a[n], {n, 0, 15}] (* G. C. Greubel, Aug 04 2022 *)

PARI

a(n) = my(ret=1); for(i=1,n, ret = ret<Kevin Ryde, Aug 13 2022

SageMath

from sage.combinat.q_analogues import q_pochhammer
def A027639(n): return (-1)^n*2^(n^2 + 2*n + 2)*q_pochhammer(n, 2, 2)
[A027639(n) for n in (0..15)] # G. C. Greubel, Aug 04 2022

Formula

a(n) = 2^(n^2 + 2*n + 2) * Product_{j=1..n} (2^j - 1).

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

a(n) ~ c * 2^((3*n^2+5*n+4)/2), where c = A048651. - Amiram Eldar, Jul 09 2025