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

1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 31, 31, 691, 42151, 29713, 505121, 642332179, 692319119, 8003636403977, 248112728523287, 593468652605200909, 50904295073459007001, 1015740532498234470066371, 701876707956280018815862361

Offset: 1

Author

Jean-François Alcover, Apr 16 2014

Keywords

• nonn
• frac

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, Chapter 16.

Links

• Robin Visser, Table of n, a(n) for n = 1..80
• Steven R. Finch, Minkowski-Siegel mass constants, January 9, 2005. [Cached copy, with permission of the author]
• Bernd C. Kellner, On asymptotic constants related to products of Bernoulli numbers and factorials, Integers 9 (2009), Article #A08, 83-106.
• John Milnor and Dale Husemoller, Symmetric Bilinear Forms, Ergeb. Math. Grenzgeb., Band 73, Springer-Verlag, New York-Heidelberg, 1973. See page 50.
• Wikipedia, Smith-Minkowski-Siegel mass formula

Crossrefs

Cf. A005134, A054909, A054911, A241122.

Programs

Mathematica

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}]

Sage

def a(n):
    if n==1: M = 1/2
    elif n%8 == 0: M = (1-2^(-n/2))*(1+2^(1-n/2))*bernoulli(n/2)/(2*factorial(n/2))
    elif n%8 in [1,7]: M = (2^((n-1)/2) + 1)/(2^n*factorial((n-1)/2))
    elif n%8 in [2,6]: M = euler_number(n/2-1)/(factorial(n/2-1)*(2^(n+1)))
    elif n%8 in [3,5]: M = (2^((n-1)/2) - 1)/(2^n*factorial((n-1)/2))
    elif n%8 == 4: M = (1-2^(-n/2))*(1-2^(1-n/2))*bernoulli(n/2)/(2*factorial(n/2))
    M *= product([abs(bernoulli(i)) for i in range(2, n, 2)])
    return abs(M).numerator()  # Robin Visser, Feb 08 2025

Formula

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