A090674 Numerators 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.

1, 23, 11237, 2482411, 272785979, 4175309343349, 525035501918789, 628141988536245979, 53917386529177385523923, 148934765720971351352763767, 428338546734334777277256756263, 6301150244751080741665843707891149, 1695881990125518108674571524660426383

Offset

1,2

Comments

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

This sequence is also related to the orbifold Euler characteristic of the non-connected graph-complex for the cyclic operad Lie, see the articles by Gerlits and by Smillie-Vogtmann (in which only the logarithm of the generating function does appear). - F. Chapoton, Feb 12 2013

Links

Table of n, a(n) for n=1..13.

M. Borinsky and K. Vogtmann, The Euler characteristic of Out(Fn), arXiv:1907.03543 [math.GR], 2019-2020; Comment. Math. Helvetici, 95 (2020), 703-748.

F. Gerlits, The Euler characteristic of graph complexes via Feynman diagrams, arXiv:math/0412094 [math.QA], 2004-2007.

C. Lanczos, A precision approximation of the Gamma Function, SIAM J. Num. Anal. B1 (1964) 86-96.

J. Smillie and K. Vogtmann, A generating function for the Euler characteristic of Out(F_n), in: Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985). J. Pure Appl. Algebra, 44(1-3):329-348, 1987.

W. Wang, Unified approaches to the approximations of the gamma function, J. Number Theory, Volume 163, June 2016, pp. 570-595.

Maple

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) ; # R. J. Mathar, Aug 28 2009

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}]; Numerator[a] (* Cantrell *)
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] = Numerator[Lanczos[n]]; A090674 = Table[ Print[a[n]]; a[n], {n, 1, nmax}] (* Jean-François Alcover, Dec 07 2011, after Peter Luschny *)

Crossrefs

Denominators are in A090675.

Sequence in context: A233213 A368142 A368145 * A013728 A028693 A324255

Adjacent sequences: A090671 A090672 A090673 * A090675 A090676 A090677

Keyword

frac,nonn

Author

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

Status

approved