cp's OEIS Frontend

A159282 Denominator of the rational coefficient in the main term in the dynamical analog of Mertens's theorem for a full n-dimensional shift, n >= 2.

%I A159282 #31 Feb 20 2021 02:38:46
%S A159282 6,12,1620,2160,2551500,3061800,33756345000,38578680000,
%T A159282 4060381958325000,4511535509250000,3168740859543387253125000,
%U A159282 3456808210410967912500000,34159303730702924635072148437500
%N A159282 Denominator of the rational coefficient in the main term in the dynamical analog of Mertens's theorem for a full n-dimensional shift, n >= 2.
%C A159282 a(n) for n >= 2 may be defined as follows. For a full n-dimensional shift, let M(N) = Sum_{L} O(L)/exp(h[L]), where the sum is over subgroups L of finite index in Z^n, O(L) is the number of points with stabilizer L, and exp(h) is the number of symbols.
%C A159282 Then M(N) is asymptotic to a rational times a power of Pi times a product of values of the zeta function at odd integers, and a(n) is the denominator of that rational.
%H A159282 Vaclav Kotesovec, <a href="/A159282/b159282.txt">Table of n, a(n) for n = 2..63</a>
%H A159282 R. Miles and T. Ward, <a href="https://doi.org/10.1090/S0002-9939-08-09649-4">Orbit-counting for nilpotent group shifts</a>, Proc. Amer. Math. Soc. 137 (2009), 1499-1507.
%F A159282 By Perron's formula, M(N) = residue(zeta(z+1) * ... * zeta(z-n+2) * N^z, z=n-1) = (b(n)/a(n)) * N^(d-1) * Pi^(floor(n/2)*(floor(n/2)+1)) * Product_{j=1..floor((n-1)/2)} zeta(2*j+1), where b(n) = A159283(n).
%e A159282 For n = 3, using the formula in terms of residues, we have residue(zeta(z-1) * zeta(z) * zeta(z+1) * N^z/z, z=2) = (1/12) * zeta(3) * Pi^2 * N^2, so a(3) = 12 (and A159283(3) = 1). [Because A159283(n) = 1 for n = 2..11, these ten values are not listed in the OEIS.]
%p A159282 # The following program generates an expression from which denominator a(n) can be read off:
%p A159282 f:=n->residue(product(Zeta(z-j),j=-1..(n-2))*N^z/z,z=n-1):
%p A159282 seq(f(n), n=2..30);
%t A159282 Denominator[Table[Residue[Product[Zeta[z - j], {j, -1, n-2}]/z, {z, n-1}], {n, 2, 14}]] (* _Vaclav Kotesovec_, Sep 05 2019 *)
%Y A159282 This is the denominator of a rational sequence whose numerator is A159283.
%K A159282 easy,frac,nonn
%O A159282 2,1
%A A159282 _Thomas Ward_, Apr 08 2009
%E A159282 Various sections edited by _Petros Hadjicostas_, Feb 20 2021