A090675 Denominators of rational coefficients in a series expansion of z! = Gamma(z+1), convergent for Re(z) >= 0, given as equation (21) in the referenced paper by Lanczos.

24, 1152, 414720, 39813120, 1337720832, 4815794995200, 115579079884800, 22191183337881600, 263631258054033408000, 88580102706155225088000, 27636992044320430227456000, 39797268543821419527536640000

Offset: 1

Author

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

Keywords

• frac
• nonn

Comments

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

It seems that for n>0 A090675(n) = A144618(n) with the exception 5*A090675(5) = A144618(5). - Peter Luschny, Mar 01 2011

References

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

Links

• W. Wang, Unified approaches to the approximations of the gamma function, J. Number Theory (2016).

Crossrefs

Numerators are in A090674.

Programs

Maple

Lanczos := proc(n)
exp(1+LambertW((x^2-1)/exp(1)));
coeftayl(taylor(%,x=0,2*n+2),x=0,2*n+1);
simplify(-%*(2*n+1)*pochhammer(1/2,n)/sqrt(2),exp) end:
A090674 := n -> numer(Lanczos(n));
A090675 := n -> denom(Lanczos(n)); # Peter Luschny, Mar 01 2011

Mathematica

(* 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}]; Denominator[a]
nmax = 10; f[x_] := Exp[1 + ProductLog[(x^2 - 1)/E]]; se = Series[f[x], {x, 0, 2 nmax + 2}] /. Arg[x] -> 0; Lanczos[n_] := ( coe = SeriesCoefficient[ se, {x, 0, 2 n + 1}]; Simplify[ -coe*(2*n + 1)*Pochhammer[1/2, n]/Sqrt[2]]); a[n_] := a[n] = Denominator[ Lanczos[n] ]; A090675 = Table[ Print[a[n]]; a[n], {n, 1, nmax}] (* Jean-François Alcover, Dec 07 2011, after Peter Luschny *)