A360809 Decimal expansion of the area under the curve of the reciprocal of the Luschny factorial function from zero to infinity.

2, 5, 8, 6, 7, 0, 5, 0, 5, 9, 7, 8, 6, 8, 0, 8, 2, 2, 7, 7, 7, 8, 1, 0, 6, 8, 7, 2, 9, 4, 6, 9, 6, 0, 2, 1, 3, 5, 7, 3, 0, 9, 6, 2, 7, 4, 2, 4, 8, 9, 3, 6, 1, 2, 4, 4, 6, 7, 0, 8, 2, 4, 2, 2, 5, 8, 5, 9, 4, 0, 4, 5, 5, 6, 0, 6, 6, 4, 3, 4, 2, 6, 4, 2, 8, 8, 2, 7, 7, 7, 5, 6, 7, 5, 3, 9, 0, 8, 8, 7, 6, 4, 4, 6, 9, 9, 8, 1

Offset

1,1

Links

Table of n, a(n) for n=1..108.

Peter Luschny, Is the Gamma-function misdefined? Or: Hadamard vs Euler - Who found the better Gamma function?.

Hywel Normington, Python Code, 2023.

Hywel Normington, Reciprocal Gamma Integrals, 2023.

Formula

L(x) = Gamma(x+1)P(x), where P(x) = 1 - g(x)*sin(Pi*x)/(Pi*x) and g(x) = (x/2)*(Psi((x+1)/2) - Psi(x/2)) - 1/2.

Equals Integral_{0..oo} 1/L(x) dx.

Example

2.58670505978680822777810687294696021357309627424893612446708242258594...

Maple

L := proc(x) local G, S, y; if x = 0 then return 0.5 fi; y := x * 0.5; if is(x < 0) then y := -y fi; G := y * (Psi(y + 0.5) - Psi(y)) - 0.5; if is(x < 0) then return G/(-x)! fi; y := Pi * x; S := sin(y) / y; (1 - S * G) * x! end: RL := x -> 1 / L(x):
IntRL := n -> evalf[n](Int(RL, 0..n, method = _Gquad)): IntRL(40); # Peter Luschny, Feb 22 2023

Mathematica

RealDigits[NIntegrate[1 / (Gamma[x+1] * (1 - (x/2 * (PolyGamma[0, (x+1)/2] - PolyGamma[0, x/2]) - 1/2) * Sin[Pi*x]/(Pi*x))), {x, 0, Infinity}, WorkingPrecision -> 110, MaxRecursion -> Infinity]][[1]] (* Vaclav Kotesovec, Feb 22 2023 *)

Prog

(PARI) default(realprecision, 500); intnum(x=0, [[1], 1], 1 / (gamma(x+1) * (1 - (x/2 * (psi((x+1)/2) - psi(x/2)) - 1/2) * sin(Pi*x)/(Pi*x)))) \\ (default(realprecision, 200) is enough for 59 valid digits, \p 500 for 102 valid digits, \p 1000 for 148 valid digits). - Vaclav Kotesovec, Feb 22 2023

Crossrefs

Cf. A360375, A058655.

Sequence in context: A360898 A118119 A340253 * A287013 A057929 A353260

Adjacent sequences: A360806 A360807 A360808 * A360810 A360811 A360812

Keyword

nonn,cons

Author

Hywel Normington, Feb 21 2023

Extensions

More digits from Vaclav Kotesovec, Feb 22 2023

Status

approved