1,2

It would be nice to have a way to generate the sequence which is simpler than that used in the program provided.

C. Lanczos, A precision approximation of the gamma function, J. SIAM Numer. Anal., Ser. B, 1 (1964), 86-96

C. Lanczos, A precision approximation of the Gamma Function, SIAM J. Num. Anal. B1 (1964) 86-96. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 28 2009]

A090674 := proc(n) local e, y, t ; e := exp(1) ; y := (x^2-1)/e ; y := e*exp(LambertW(y)) ; taylor(y, x=0, 2*n+2) ; simplify(coeftayl(%, x=0, 2*n+1), exp) ; %*doublefactorial(2*n+1)/2^n/sqrt(2) ; abs(numer(%)) ; end: seq( A090674(n), n=1..14) ; [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 28 2009]

(* Gamma[z+1] == Sqrt[2*Pi]*((z + 1/2)/E)^(z + 1/2)*(1 - Sum[a[[n]]/Pochhammer[z + 1, n], {n, 1, Infinity}] *) n = 30 (* which must be even *); e[0] = 1; e[1] = Sqrt[2]; f[x_] := SeriesData[x, 0, Table[e[i], {i, 0, n}], 0, n + 1, 1]; d = First[Table[e[i], {i, 0, n - 1}] /. Solve[CoefficientList[Normal[(1/2)*D[f[x]^2, x] - (1 - x^2)*D[f[x], x] - 2*x*f[x]], x] == 0, Table[e[i], {i, 2, n}]]]; c = Table[Sqrt[2]*(i - 1)*d[[i]]*Sin[theta]^(i - 2), {i, 2, n, 2}]; b = Table[Integrate[Cos[theta]^(2*x)*c[[i]], {theta, -(Pi/2), Pi/2}, Assumptions -> x > -(1/2)], {i, 1, n/2}]; a = Table[ -((b[[i]]*Gamma[i + x])/(2*Sqrt[Pi]*Gamma[1/2 + x])), {i, 2, n/2}]; Numerator[a]

Denominators are in A090675.

Sequence in context: A134801 A059932 A045735 this_sequence A013728 A028693 A033998

Adjacent sequences: A090671 A090672 A090673 this_sequence A090675 A090676 A090677

frac,nonn

David W. Cantrell (DWCantrell(AT)sigmaxi.net), Dec 18 2003