A143503 Numerators in the asymptotic expansion of Gamma(x+1/2)/Gamma(x).
1, -1, 1, 5, -21, -399, 869, 39325, -334477, -28717403, 59697183, 8400372435, -34429291905, -7199255611995, 14631594576045, 4251206967062925, -68787420596367165, -26475975382085110035, 53392138323683746235, 26275374869163335461975, -105772979046693606062363
Offset: 1
Examples
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 + ...
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.]
- D. H. Lehmer, Interesting series involving the central binomial coefficient, Amer. Math. Monthly, 92(7) (1985), 449-457.
- Eric Weisstein's World of Mathematics, Gamma Function.
Programs
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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); seq(A143503(n), n=1..16); # Peter Luschny, Apr 05 2014
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Mathematica
Numerator[CoefficientList[Series[Gamma[x + 1/2]/Gamma[x]/Sqrt[x], {x, Infinity, 20}], 1/x]] (* Vaclav Kotesovec, Oct 09 2023 *)
Extensions
More terms from Vaclav Kotesovec, Oct 09 2023