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A007788 Number of augmented Andre 3-signed permutations: E.g.f. (1-sin(3*x))^(-1/3).

%I A007788 #44 Jun 24 2025 10:53:59
%S A007788 1,1,4,19,136,1201,13024,165619,2425216,40132801,740882944,
%T A007788 15091932019,336257744896,8134269015601,212309523595264,
%U A007788 5946914908771219,177934946000306176,5663754614516217601,191097349696090537984,6812679868133940475219,255885704427935576621056
%N A007788 Number of augmented Andre 3-signed permutations: E.g.f. (1-sin(3*x))^(-1/3).
%C A007788 It appears that all members are of the form 3k+1. - _Ralf Stephan_, Nov 12 2007
%H A007788 Vincenzo Librandi, <a href="/A007788/b007788.txt">Table of n, a(n) for n = 0..200</a>
%H A007788 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 A007788 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 A007788 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 A007788 E.g.f.: (1-sin(3*x))^(-1/3).
%F A007788 a(n) ~ n! * 2*6^n/(Pi^(n+2/3)*n^(1/3)*Gamma(2/3)). - _Vaclav Kotesovec_, Jun 25 2013
%F A007788 a(n) = Sum_{k=0..n} A007559(k) * (3*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - _Seiichi Manyama_, Jun 24 2025
%p A007788 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 A007788 With[{nn=20},CoefficientList[Series[(1-Sin[3x])^(-1/3),{x,0,nn}], x] Range[0,nn]!] (* _Harvey P. Dale_, Nov 23 2011 *)
%o A007788 (Magma) R<x>:=PowerSeriesRing(Rationals(), 20); Coefficients(R!(Laplace( (1-Sin(3*x))^(-1/3) ))); // _G. C. Greubel_, Mar 05 2020
%o A007788 (PARI) Vec(serlaplace( (1-sin(3*x))^(-1/3) +O('x^20) )) \\ _G. C. Greubel_, Mar 05 2020
%o A007788 (PARI) a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
%o A007788 a007559(n) = prod(k=0, n-1, 3*k+1);
%o A007788 a(n) = sum(k=0, n, a007559(k)*(3*I)^(n-k)*a136630(n, k)); \\ _Seiichi Manyama_, Jun 24 2025
%o A007788 (Sage)
%o A007788 m=20;
%o A007788 def A007788_list(prec):
%o A007788     P.<x> = PowerSeriesRing(QQ, prec)
%o A007788     return P( (1-sin(3*x))^(-1/3) ).list()
%o A007788 a=A007788_list(m+1); [factorial(n)*a[n] for n in (0..m)] # _G. C. Greubel_, Mar 05 2020
%Y A007788 Cf. A001586, A144015, A230134, A227544, A235128, A230114.
%Y A007788 Cf. A007863, A235135, A235132.
%Y A007788 Cf. A007559, A136630.
%K A007788 nonn
%O A007788 0,3
%A A007788 R. Ehrenborg (ehrenbor(AT)lacim.uqam.ca) and M. A. Readdy (readdy(AT)lacim.uqam.ca)