A292754 Numerators of coefficients in an asymptotic expansion of the Wallis sequence in inverse powers of n.

1, -1, 5, -11, 83, -143, 625, -1843, 24323, 61477, -14165, -8084893, 31181719, 1682401061, -3166220215, -251783137859, 3865962456803, 394670372519917, -765052915887545, -98394908192751193, 384080734825119709, 60838795345430052431, -119312155199695296505, -22845758944383820991909

Offset: 0

Author

N. J. A. Sloane, Sep 25 2017

Keywords

• sign

References

• Chao-Ping Chen, Richard B. Paris, On the asymptotic expansions of products related to the Wallis, Weierstrass, and Wilf formulas, Applied Mathematics and Computation 293 (2017) 30-39. See (3.12).

Links

• Chao-Ping Chen, Richard B. Paris, On the asymptotic expansions of products related to the Wallis, Weierstrass, and Wilf formulas, arXiv:1511.09217 [math.CA], 2015. See (3.12).

Crossrefs

Cf. A088802 or A123854 (denominators).

Programs

Mathematica

nu[j_] := (-1)^(j+1) ((4 - 2^(1-j)) BernoulliB[j+1] - (j+1) 2^(-j))/(j*(j + 1)); mu[j_] := mu[j] = If[j == 0, 1, Sum[k nu[k] mu[j-k], {k, j}]/j]; Table[Numerator@mu@n, {n, 0, 23}] (* Giovanni Resta, May 29 2019 *)
Numerator[CoefficientList[Series[16^n/(Pi*(2*n + 1) * Binomial[2*n, n]^2), {n, Infinity, 20}], 1/n]] (* Vaclav Kotesovec, Jun 02 2019 *)

PARI

nu(j) = (-1)^(j+1)*((4-2^(1-j))*bernfrac(j+1) - (j+1)*2^(-j))/(j*(j+1));
mu(j) = if (j==0, 1, sum(k=1, j, k*nu(k)*mu(j-k))/j);
a(n) = numerator(mu(n)); \\ Michel Marcus, May 29 2019

Formula

See (3.8) and (3.11) in Chen link.

Extensions

More terms from Michel Marcus, May 29 2019