A007289 Expansion of e.g.f. (sin(2*x) + cos(x)) / cos(3*x).

1, 2, 8, 46, 352, 3362, 38528, 515086, 7869952, 135274562, 2583554048, 54276473326, 1243925143552, 30884386347362, 825787662368768, 23657073914466766, 722906928498737152, 23471059057478981762, 806875574817679474688, 29279357851856595135406

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

• Matthew House, Table of n, a(n) for n = 0..406 (terms 0..200 from Vincenzo Librandi)
• 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 (1995), 173-196.

Crossrefs

Row 3 of A349271.

Cf. A006873, A007286, A225109, A000191 (bisection), A000436 (bisection).

Cf. A000111, A136630, A352638.

Programs

Maple

A007289 := proc(n) local k,j; add(add((-1)^j*binomial(k,j)*(k-2*j)^n*I^(n-k),j=0..k),k=0..n) end: # Peter Luschny, Jul 31 2011

Mathematica

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

PARI

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

Sage

from mpmath import mp, polylog, im
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 A007289(n) : return im(aperm3(n))
[int(A007289(n)) for n in (0..17)] # Peter Luschny, Apr 28 2013

Formula

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

a(n) = Sum_{k=0..n} Sum_{j=0..k} (-1)^j*binomial(k,j)*(k-2*j)^n*I^(n-k). - Peter Luschny, Jul 31 2011

a(n) = Im(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+1/2)/Pi^(n+1). - Vaclav Kotesovec, Jun 15 2013

a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} (-1)^k * binomial(n,2*k+1) * a(n-2*k-1). - Ilya Gutkovskiy, Mar 10 2022

From Seiichi Manyama, Jun 25 2025: (Start)

E.g.f.: 1/(1 - 2 * sin(x)).

a(n) = Sum_{k=0..n} 2^k * k! * i^(n-k) * A136630(n,k), where i is the imaginary unit. (End)