A143503 - OEIS (original) (raw)

A143503

Numerators in the asymptotic expansion of Gamma(x+1/2)/Gamma(x).

2

1, -1, 1, 5, -21, -399, 869, 39325, -334477, -28717403, 59697183, 8400372435, -34429291905, -7199255611995, 14631594576045, 4251206967062925, -68787420596367165, -26475975382085110035, 53392138323683746235, 26275374869163335461975, -105772979046693606062363

LINKS

F. J. Dyson, N. E. Frankel and M. L. Glasser, Lehmer's Interesting Series, arXiv:1009.4274 [math-ph], 2010-2011. [Except for the signs, see the unnumbered table on p. 7.]

F. J. Dyson, N. E. Frankel and M. L. Glasser, Lehmer's interesting series, Amer. Math. Monthly, 120 (2013), 116-130. [Except for the signs, see Table 4.]

EXAMPLE

1/sqrt(x^(-1)) - sqrt(x^(-1))/8 + (x^(-1))^(3/2)/128 + (5*(x^(-1))^(5/2))/1024 - (21*(x^(-1))^(7/2))/32768 + ...

MAPLE

H := proc(n) local S, i; S := (x/(exp(x)-1))^(3/2)*exp(x/2);

-pochhammer(1/2, n-1)*coeff(series(S, x, n+2), x, n)*2^(4*n-1-add(i, i= convert(n, base, 2))) end:

A143503 := n -> (-1)^irem(n-1, 6)*H(n-1);

MATHEMATICA

Numerator[CoefficientList[Series[Gamma[x + 1/2]/Gamma[x]/Sqrt[x], {x, Infinity, 20}], 1/x]] (* Vaclav Kotesovec, Oct 09 2023 *)