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

1, 1, 4, 19, 136, 1201, 13024, 165619, 2425216, 40132801, 740882944, 15091932019, 336257744896, 8134269015601, 212309523595264, 5946914908771219, 177934946000306176, 5663754614516217601, 191097349696090537984, 6812679868133940475219, 255885704427935576621056

Offset: 0

Author

R. Ehrenborg (ehrenbor(AT)lacim.uqam.ca) and M. A. Readdy (readdy(AT)lacim.uqam.ca)

Keywords

• nonn

Comments

It appears that all members are of the form 3k+1. - Ralf Stephan, Nov 12 2007

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• R. Ehrenborg and M. A. Readdy, Sheffer posets and r-signed permutations, Preprint submitted to Ann. Sci. Math. Quebec, 1994. (Annotated scanned copy)
• R. Ehrenborg and M. A. Readdy, Sheffer posets and r-signed permutations, Ann. Sci. Math. Québec, 19 (1995), no. 2, 173-196.
• R. Ehrenborg and M. A. Readdy, The r-cubical lattice and a generalization of the cd-index, European J. Combin. 17 (1996), no. 8, 709-725.

Crossrefs

Cf. A001586, A144015, A230134, A227544, A235128, A230114.

Cf. A007863, A235135, A235132.

Cf. A007559, A136630.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!(Laplace( (1-Sin(3*x))^(-1/3) ))); // G. C. Greubel, Mar 05 2020

Maple

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

Mathematica

With[{nn=20},CoefficientList[Series[(1-Sin[3x])^(-1/3),{x,0,nn}], x] Range[0,nn]!] (* Harvey P. Dale, Nov 23 2011 *)

PARI

Vec(serlaplace( (1-sin(3*x))^(-1/3) +O('x^20) )) \\ G. C. Greubel, Mar 05 2020

PARI

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*(3*I)^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

Sage

m=20;
def A007788_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( (1-sin(3*x))^(-1/3) ).list()
a=A007788_list(m+1); [factorial(n)*a[n] for n in (0..m)] # G. C. Greubel, Mar 05 2020

Formula

E.g.f.: (1-sin(3*x))^(-1/3).

a(n) ~ n! * 2*6^n/(Pi^(n+2/3)*n^(1/3)*Gamma(2/3)). - Vaclav Kotesovec, Jun 25 2013

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