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%I #38 Apr 19 2021 08:04:59
%S 1,23,11237,2482411,272785979,4175309343349,525035501918789,
%T 628141988536245979,53917386529177385523923,
%U 148934765720971351352763767,428338546734334777277256756263,6301150244751080741665843707891149,1695881990125518108674571524660426383
%N 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.
%C It would be nice to have a way to generate the sequence which is simpler than that used in the [first Mathematica] program provided.
%C 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
%H M. Borinsky and K. Vogtmann, <a href="https://arxiv.org/abs/1907.03543">The Euler characteristic of Out(Fn)</a>, arXiv:1907.03543 [math.GR], 2019-2020; Comment. Math. Helvetici, 95 (2020), 703-748.
%H F. Gerlits, <a href="http://arxiv.org/abs/math/0412094">The Euler characteristic of graph complexes via Feynman diagrams</a>, arXiv:math/0412094 [math.QA], 2004-2007.
%H C. Lanczos, <a href="http://dx.doi.org/10.1137/0701008">A precision approximation of the Gamma Function</a>, SIAM J. Num. Anal. B1 (1964) 86-96.
%H J. Smillie and K. Vogtmann, <a href="http://dx.doi.org/10.1016/0022-4049(87)90036-3">A generating function for the Euler characteristic of Out(F_n)</a>, in: Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985). J. Pure Appl. Algebra, 44(1-3):329-348, 1987.
%H W. Wang, <a href="http://dx.doi.org/10.1016/j.jnt.2015.12.016">Unified approaches to the approximations of the gamma function</a>, J. Number Theory, Volume 163, June 2016, pp. 570-595.
%p 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
%t (* 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 *)
%t 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_ *)
%Y Denominators are in A090675.
%K frac,nonn
%O 1,2
%A David W. Cantrell (DWCantrell(AT)sigmaxi.net), Dec 18 2003
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