A340471 Denominators of an approximation to zeta(n)/Pi^n.

2, 6, 28, 90, 1488, 945, 182880, 9450, 8241408, 93555, 14856307200, 638512875, 1569400842240, 18243225, 5713142135500800, 325641566250, 1096948397364019200, 38979295480125, 6713362606110031872000, 1531329465290625, 408173030347971900211200, 13447856940643125

Offset: 1

Author

Melchor Viso Martinez, Jan 08 2021

Keywords

• nonn
• frac

Examples

			1/2, 1/6, 1/28, 1/90, 5/1488, 1/945, 61/182880, 1/9450, 277/8241408, 1/93555, 50521/14856307200, 691/638512875, ...
Values are approximate for odd indices, exact for even indices:
  zeta(1) ~       1/2             zeta(2) = Pi^2/6
  zeta(3) ~    Pi^3/28            zeta(4) = Pi^4/90
  zeta(5) ~  5*Pi^5/1488          zeta(6) = Pi^6/945
  zeta(7) ~ 61*Pi^7/182880,       zeta(8) = Pi^8/9450
  ...
		

Links

• Melchor Viso Martinez, An expression for integer zeta approximation

Crossrefs

Cf. A000831, A002432, A331839, A340472 (numerators).

Programs

Mathematica

a[k_] := Denominator[(1/(4 (1 - 2^-k) k!)
      D[\[Lambda] Tan[(\[Pi] + \[Lambda])/4], {\[Lambda],
       k}]) /. {\[Lambda] -> 0}]

PARI

a(n) = {my(t=tan(x/4 + O(x*x^n))); denominator(polcoef(x*(1 + t)/(1 - t), n)/((4-2^(2-n))))} \\ Andrew Howroyd, Jan 10 2021

Formula

a(n) = denominator of lim_{x->0} of the n-th derivative of x*tan((Pi+x)/4)/((4-2^(2-n))*n!) with respect to x.

a(2*n) = A002432(n).

From Andrew Howroyd, Jan 10 2021: (Start)

a(n) = denominator of (1/(4-2^(2-n)))*[x^n] x*(1 + tan(x/4))/(1 - tan(x/4)).

a(n) = denominator( A000831(n-1)/((n-1)!*2^n*(2^n-1)) ). (End)