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A007289 E.g.f.: (sin 2x + cos x) / cos 3x.
(Formerly M1873)
6
1, 2, 8, 46, 352, 3362, 38528, 515086, 7869952, 135274562, 2583554048, 54276473326, 1243925143552, 30884386347362, 825787662368768, 23657073914466766, 722906928498737152, 23471059057478981762 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

Richard Ehrenborg and Margaret A. Readdy, Sheffer posets and r-signed permutations, Annales des Sciences Mathématiques du Québec {\bf 19} (1995), 173-196.

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

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 *)

PROG

(PARI) 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.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))

[A007289(n) for n in (0..17)] # Peter Luschny, Apr 28 2013

CROSSREFS

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

Sequence in context: A052801 A180390 A074599 * A099765 A246386 A096656

Adjacent sequences:  A007286 A007287 A007288 * A007290 A007291 A007292

KEYWORD

nonn

AUTHOR

N. J. A. Sloane and Simon Plouffe

STATUS

approved

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Last modified March 28 12:08 EDT 2017. Contains 284186 sequences.