A340472 Numerators of an approximation to zeta(n)/Pi^n.

1, 1, 1, 1, 5, 1, 61, 1, 277, 1, 50521, 691, 540553, 2, 199360981, 3617, 3878302429, 43867, 2404879675441, 174611, 14814847529501, 155366, 69348874393137901, 236364091, 238685140977801337, 1315862, 4087072509293123892361, 6785560294, 13181680435827682794403, 6892673020804

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. A046988, A340471 (denominators).

Programs

Mathematica

a[k_] := Numerator[(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))); numerator(polcoef(x*(1 + t)/(1 - t), n)/((4-2^(2-n))))} \\ Andrew Howroyd, Jan 10 2021

Formula

a(n) = numerator 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) = A046988(n).