A249185 Decimal expansion of a constant appearing in the Hankel determinant asymptotics.
6, 4, 5, 0, 0, 2, 4, 4, 8, 5, 0, 9, 5, 7, 7, 0, 8, 4, 6, 5, 8, 9, 6, 1, 0, 0, 7, 7, 2, 1, 7, 8, 7, 6, 5, 5, 3, 4, 7, 6, 1, 4, 4, 9, 4, 0, 5, 7, 3, 3, 9, 7, 2, 1, 5, 5, 2, 1, 4, 4, 5, 8, 8, 5, 8, 0, 2, 7, 6, 0, 7, 8, 7, 4, 1, 2, 4, 6, 8, 4, 6, 5, 7, 3, 9, 7, 1, 0, 5, 4, 9, 7, 1, 9, 7, 4, 0, 9, 9, 1, 4, 6
Offset: 0
Examples
0.645002448509577084658961007721787655347614494...
Links
- Steven Finch, Hankel and Toeplitz Determinants, March 17, 2014. [Cached copy, with permission of the author]
- Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 416.
- Eric Weisstein's MathWorld, Hilbert matrix
- Wikipedia, Hilbert matrix
Programs
-
Maple
evalf(limit(2^(1/12) * n^(3*n^2/2 + 3*n/2 + 1/4) * exp(1/4-3*n^2/4) / product(k^(3*k), k=1..n), n=infinity),120); # Vaclav Kotesovec, Oct 23 2014
-
Mathematica
h = 2^(1/12)*E^(1/4)*Glaisher^-3; RealDigits[h, 10, 102] // First
Formula
Det(H_n) ~ h*4^(-n^2)*(2*Pi)^n*n^(-1/4), where h = 2^(1/12)*e^(1/4)*A^(-3), A denoting the Glaisher-Kinkelin constant.