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A241121 Type I Minkowski-Siegel mass constants (numerators).

%I A241121 #20 Feb 10 2025 02:10:58
%S A241121 1,1,1,1,1,1,1,1,17,1,31,31,691,42151,29713,505121,642332179,
%T A241121 692319119,8003636403977,248112728523287,593468652605200909,
%U A241121 50904295073459007001,1015740532498234470066371,701876707956280018815862361
%N A241121 Type I Minkowski-Siegel mass constants (numerators).
%D A241121 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, Chapter 16.
%H A241121 Robin Visser, <a href="/A241121/b241121.txt">Table of n, a(n) for n = 1..80</a>
%H A241121 Steven R. Finch, <a href="/A241121/a241121.pdf">Minkowski-Siegel mass constants</a>, January 9, 2005. [Cached copy, with permission of the author]
%H A241121 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 A241121 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 A241121 Wikipedia, <a href="http://en.wikipedia.org/wiki/Smith%E2%80%93Minkowski%E2%80%93Siegel_mass_formula">Smith-Minkowski-Siegel mass formula</a>
%F A241121 a(n)/A241122(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 A241121 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] // Numerator, {n, 1, 30}]
%o A241121 (Sage)
%o A241121 def a(n):
%o A241121     if n==1: M = 1/2
%o A241121     elif n%8 == 0: M = (1-2^(-n/2))*(1+2^(1-n/2))*bernoulli(n/2)/(2*factorial(n/2))
%o A241121     elif n%8 in [1,7]: M = (2^((n-1)/2) + 1)/(2^n*factorial((n-1)/2))
%o A241121     elif n%8 in [2,6]: M = euler_number(n/2-1)/(factorial(n/2-1)*(2^(n+1)))
%o A241121     elif n%8 in [3,5]: M = (2^((n-1)/2) - 1)/(2^n*factorial((n-1)/2))
%o A241121     elif n%8 == 4: M = (1-2^(-n/2))*(1-2^(1-n/2))*bernoulli(n/2)/(2*factorial(n/2))
%o A241121     M *= product([abs(bernoulli(i)) for i in range(2, n, 2)])
%o A241121     return abs(M).numerator()  # _Robin Visser_, Feb 08 2025
%Y A241121 Cf. A005134, A054909, A054911, A241122.
%K A241121 nonn,frac
%O A241121 1,9
%A A241121 _Jean-François Alcover_, Apr 16 2014