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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. 3
1, 23, 11237, 2482411, 272785979, 4175309343349, 525035501918789, 628141988536245979, 53917386529177385523923, 148934765720971351352763767, 428338546734334777277256756263, 6301150244751080741665843707891149, 1695881990125518108674571524660426383 (list; graph; refs; listen; history; text; internal format)
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.

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

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 (2016).

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-Fran├žois Alcover, Dec 07 2011, after Peter Luschny *)

CROSSREFS

Denominators are in A090675.

Sequence in context: A233109 A193430 A233213 * A013728 A028693 A273940

Adjacent sequences:  A090671 A090672 A090673 * A090675 A090676 A090677

KEYWORD

frac,nonn

AUTHOR

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

STATUS

approved

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Last modified September 22 00:25 EDT 2017. Contains 292326 sequences.