A007286 E.g.f.: (sin x + cos 2x) / cos 3x.

1, 1, 5, 26, 205, 1936, 22265, 297296, 4544185, 78098176, 1491632525, 31336418816, 718181418565, 17831101321216, 476768795646785, 13658417358350336, 417370516232719345, 13551022195053101056

Offset: 0

Author

N. J. A. Sloane and Simon Plouffe

Keywords

• nonn

Comments

Arises in the enumeration of alternating 3-signed permutations.

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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, 1994. (Annotated scanned copy)
• Richard Ehrenborg and Margaret A. Readdy, Sheffer posets and r-signed permutations, Annales des Sciences Mathématiques du Québec 19:2 (1995), 173-196.

Crossrefs

Cf. A006873, A007289, A225109, A002438 (bisection?).

Programs

Mathematica

mx = 17; Range[0, mx]! CoefficientList[ Series[ (Sin[x] + Cos[2x])/Cos[3 x], {x, 0, mx}], x] (* Robert G. Wilson v, Apr 28 2013 *)

PARI

x='x+O('x^66); Vec(serlaplace((sin(x)+cos(2*x))/cos(3*x))) \\ Joerg Arndt, Apr 28 2013

Sage

from mpmath import mp, polylog, re
mp.dps = 32; mp.pretty = True
def aperm3(n): return 2*((1-I)/(1+I))^n*(1+add(binomial(n,j)*polylog(-j,I)*3^j for j in (0..n)))
def A007286(n) : return re(aperm3(n))
[int(A007286(n)) for n in (0..17)] # Peter Luschny, Apr 28 2013

Formula

a(n) = Re(2*((1-I)/(1+I))^n*(1+sum_{j=0..n}(binomial(n,j)*Li_{-j}(I)*3^j))). - Peter Luschny, Apr 28 2013

a(n) ~ n! * 2^(n+1)*3^n/Pi^(n+1). - Vaclav Kotesovec, Jun 15 2013