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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. 2
6, 12, 1620, 2160, 2551500, 3061800, 33756345000, 38578680000, 4060381958325000, 4511535509250000, 3168740859543387253125000, 3456808210410967912500000, 34159303730702924635072148437500 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

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.

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.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 2..63

R. Miles and T. Ward, Orbit-counting for nilpotent group shifts, Proc. Amer. Math. Soc. 137 (2009), 1499-1507.

FORMULA

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).

EXAMPLE

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.]

MAPLE

# The following program generates an expression from which denominator a(n) can be read off:

f:=n->residue(product(Zeta(z-j), j=-1..(n-2))*N^z/z, z=n-1):

seq(f(n), n=2..30);

MATHEMATICA

Denominator[Table[Residue[Product[Zeta[z - j], {j, -1, n-2}]/z, {z, n-1}], {n, 2, 14}]] (* Vaclav Kotesovec, Sep 05 2019 *)

CROSSREFS

This is the denominator of a rational sequence whose numerator is A159283.

Sequence in context: A259130 A032511 A036900 * A202383 A216423 A229336

Adjacent sequences:  A159279 A159280 A159281 * A159283 A159284 A159285

KEYWORD

easy,frac,nonn,changed

AUTHOR

Thomas Ward, Apr 08 2009

EXTENSIONS

Various sections edited by Petros Hadjicostas, Feb 20 2021

STATUS

approved

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Last modified February 28 05:24 EST 2021. Contains 341695 sequences. (Running on oeis4.)