A215416 Numerator of zeta function multiple arising in Zig-Zag Conjecture.

12, 2, 9, 155, 903, 10731, 67617, 3513939, 23429835, 318633601, 2201489511, 61641559343, 436232833827, 6231896697475, 44869657893345, 5204880267153795, 37965009095868915, 556820133081965445, 4102885191727320075

Offset

1,1

Comments

The period of the zig-zag graph Z_n = a(n)*Zeta(2*n-3) where Zeta is the Riemann zeta function. For n = 1, 2, 3... the rational multiples of the zeta function are 12/1, 2/1, 9/1, 155/8, 903/16, 10731/64, 67617/128, 3513939/2048, 23429835/4096, 318633601/16384.

Links

Table of n, a(n) for n=1..19.

Francis Brown and Oliver Schnetz, Proof of the zig-zag conjecture, arXiv:1208.1890 [math.NT], 2012-2013.

Formula

a(n) = 4*((2*n-2)!)/(n!* (n-1)!)*(1 - ((-1)^n)/(2^(2*n-3))).

Example

a(1) = 12 = numerator of 4*((2*1-2)!)/(1!* (1-1)!)*(1 - ((-1)^1)/(2^(2*1-3))) = 12/1.
a(2) = 2 = numerator of = 4*((2*2-2)!)/(2!* (2-1)!)*(1 - ((-1)^2)/(2^(2*2-3))) = 2/1.
a(3) = 9 = numerator of 4*((2*3-2)!)/(3!* (3-1)!)*(1 - ((-1)^3)/(2^(2*3-3))) = 9/1.
a(4) = 155 = numerator of 4*((2*4-2)!)/(4!* (4-1)!)*(1 - ((-1)^4)/(2^(2*4-3))) = 155/8.

Maple

A215416 := proc(n)
    4* (2*n-2)!/n!/(n-1)!*(1-(-1)^n/2^(2*n-3)) ;
    numer(%) ;
end proc: # R. J. Mathar, Aug 10 2012

Mathematica

a[n_] := 4 (2*n - 2)!/n!/(n - 1)!*(1 - (-1)^n/2^(2*n - 3)) // Numerator;
Array[a, 20] (* Jean-François Alcover, Dec 02 2017 *)

Prog

(Sage)
[(4*binomial(2*n-2, n-1)/n*(1-(-1)**n*2**(-2*n+3))).numerator() for n in range(1, 20)] # F. Chapoton, Mar 03 2020

Crossrefs

Sequence in context: A107832 A322521 A099136 * A264970 A287205 A183729

Adjacent sequences: A215413 A215414 A215415 * A215417 A215418 A215419

Keyword

nonn,easy

Author

Jonathan Vos Post, Aug 09 2012

Status

approved