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

1,1

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

Table of n, a(n) for n=1..12.

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

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 *)

Crossrefs

Numerators are in A090674.

Sequence in context: A080775 A191744 A323994 * A144618 A042107 A042104

Adjacent sequences: A090672 A090673 A090674 * A090676 A090677 A090678

Keyword

frac,nonn

Author

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

Status

approved