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A241122 Type I Minkowski-Siegel mass constants (denominators).

%I A241122 #17 Feb 10 2025 02:08:39
%S A241122 2,8,48,384,3840,46080,645120,10321920,2786918400,2229534720,
%T A241122 735746457600,5885971660800,765176315904000,192824431607808000,
%U A241122 385648863215616000,12340763622899712000,18881368343036559360000,15105094674429247488000
%N A241122 Type I Minkowski-Siegel mass constants (denominators).
%D A241122 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, Chapter 16.
%H A241122 Robin Visser, <a href="/A241122/b241122.txt">Table of n, a(n) for n = 1..400</a>
%H A241122 Steven R. Finch, <a href="/A241121/a241121.pdf">Minkowski-Siegel mass constants</a>, January 9, 2005. [Cached copy, with permission of the author]
%H A241122 Bernd C. Kellner, <a href="https://doi.org/10.1515/INTEG.2009.009">On asymptotic constants related to products of Bernoulli numbers and factorials</a>, Integers 9 (2009), Article #A08, 83-106.
%H A241122 John Milnor and Dale Husemoller, <a href="https://doi.org/10.1007/978-3-642-88330-9">Symmetric Bilinear Forms</a>, Ergeb. Math. Grenzgeb., Band 73, Springer-Verlag, New York-Heidelberg, 1973.  See page 50.
%H A241122 Wikipedia, <a href="http://en.wikipedia.org/wiki/Smith%E2%80%93Minkowski%E2%80%93Siegel_mass_formula">Smith-Minkowski-Siegel mass formula</a>
%F A241122 A241121(n)/a(n) ~ C * (n/(2*Pi*e*sqrt(e)))^(n^2/4) * (8*Pi*e/n)^(n/4) * (1/n)^(1/24), where C = 2^(-5/4) * e^(1/24) * exp(1/12 - zeta'(-1))^(-1/2) * Product_{i>=1} zeta(2*i) = 0.7048648734... [Kellner and Milnor--Husemoller]. - _Robin Visser_, Feb 08 2025
%t A241122 a[n_ /; 1 <= n <= 8] = 1/(n!*2^n); a[n_ /; n > 8] := (k = Quotient[n, 2]; r = Mod[n, 8]; Switch[r, 0, (1 - 2^-k)*(1 + 2^(1-k))/(k!*2)*BernoulliB[k]*Product[BernoulliB[j], {j, 2, 2k-2, 2}], 1|7, (2^k+1)/(k!*2^(2k+1))*Product[BernoulliB[j], {j, 2, 2k, 2}], 2|6, 1/((k-1)!*2^(2k+1))*EulerE[k-1]*Product[BernoulliB[j], {j, 2, 2k-2, 2}],3|5, (2^k-1)/(k!*2^(2k+1))*Product[BernoulliB[j], {j, 2, 2k, 2}], 4, (1-2^-k)*(1-2^(1-k))/(k!*2)*BernoulliB[k]* Product[BernoulliB[j], {j, 2, 2k-2, 2}], _, Print["error n = ", n]; 0] // Abs); Table[a[n] // Denominator, {n, 1, 30}]
%o A241122 (Sage)
%o A241122 def a(n):
%o A241122     if n==1: M = 1/2
%o A241122     elif n%8 == 0: M = (1-2^(-n/2))*(1+2^(1-n/2))*bernoulli(n/2)/(2*factorial(n/2))
%o A241122     elif n%8 in [1, 7]: M = (2^((n-1)/2) + 1)/(2^n*factorial((n-1)/2))
%o A241122     elif n%8 in [2, 6]: M = euler_number(n/2-1)/(factorial(n/2-1)*(2^(n+1)))
%o A241122     elif n%8 in [3, 5]: M = (2^((n-1)/2) - 1)/(2^n*factorial((n-1)/2))
%o A241122     elif n%8 == 4: M = (1-2^(-n/2))*(1-2^(1-n/2))*bernoulli(n/2)/(2*factorial(n/2))
%o A241122     M *= product([abs(bernoulli(i)) for i in range(2, n, 2)])
%o A241122     return abs(M).denominator()  # _Robin Visser_, Feb 08 2025
%Y A241122 Cf. A000165, A005134, A054909, A054911, A241121.
%K A241122 nonn,frac
%O A241122 1,1
%A A241122 _Jean-François Alcover_, Apr 16 2014