%I #31 Sep 08 2022 08:44:35
%S 1,1,4,19,136,1201,13024,165619,2425216,40132801,740882944,
%T 15091932019,336257744896,8134269015601,212309523595264,
%U 5946914908771219,177934946000306176,5663754614516217601
%N Number of augmented Andre 3-signed permutations: E.g.f. (1-sin(3*x))^(-1/3).
%C It appears that all members are of the form 3k+1. - _Ralf Stephan_, Nov 12 2007
%H Vincenzo Librandi, <a href="/A007788/b007788.txt">Table of n, a(n) for n = 0..200</a>
%H R. Ehrenborg and M. A. Readdy, <a href="/A007788/a007788.pdf">Sheffer posets and r-signed permutations</a>, Preprint submitted to Ann. Sci. Math. Quebec, 1994. (Annotated scanned copy)
%H R. Ehrenborg and M. A. Readdy, <a href="http://www.labmath.uqam.ca/~annales/volumes/19-2/PDF/173-196.pdf">Sheffer posets and r-signed permutations</a>, Ann. Sci. Math. Québec, 19 (1995), no. 2, 173-196.
%H R. Ehrenborg and M. A. Readdy, <a href="https://doi.org/10.1006/eujc.1996.0062">The r-cubical lattice and a generalization of the cd-index</a>, European J. Combin. 17 (1996), no. 8, 709-725.
%F E.g.f.: (1-sin(3*x))^(-1/3).
%F a(n) ~ n! * 2*6^n/(Pi^(n+2/3)*n^(1/3)*Gamma(2/3)). - _Vaclav Kotesovec_, Jun 25 2013
%p m:=20; S:=series( (1-sin(3*x))^(-1/3), x, m+1): seq(j!*coeff(S, x, j), j=0..m); # _G. C. Greubel_, Mar 05 2020
%t With[{nn=20},CoefficientList[Series[(1-Sin[3x])^(-1/3),{x,0,nn}], x] Range[0,nn]!] (* _Harvey P. Dale_, Nov 23 2011 *)
%o (PARI) Vec(serlaplace( (1-sin(3*x))^(-1/3) +O('x^20) )) \\ _G. C. Greubel_, Mar 05 2020
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 20); Coefficients(R!(Laplace( (1-Sin(3*x))^(-1/3) ))); // _G. C. Greubel_, Mar 05 2020
%o (Sage)
%o m=20;
%o def A007788_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P( (1-sin(3*x))^(-1/3) ).list()
%o a=A007788_list(m+1); [factorial(n)*a[n] for n in (0..m)] # _G. C. Greubel_, Mar 05 2020
%Y Cf. A007863, A235135, A235132.
%K nonn
%O 0,3
%A R. Ehrenborg (ehrenbor(AT)lacim.uqam.ca) and M. A. Readdy (readdy(AT)lacim.uqam.ca)