A006873 Number of alternating 4-signed permutations. (Formerly M4430)

1, 1, 7, 47, 497, 6241, 95767, 1704527, 34741217, 796079041, 20273087527, 567864586607, 17352768515537, 574448847467041, 20479521468959287, 782259922208550287, 31872138933891307457, 1379749466246228538241, 63243057486503656319047, 3059895336952604166395567

Offset

0,3

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 (1995), 173-196.

Formula

E.g.f.: (sin(x) + cos(3*x)) / cos(4*x). - M. F. Hasler, Apr 28 2013

a(n) = Re(2*((1-I)/(1+I))^n*(1 + Sum_{j=0..n}(binomial(n,j)*Li_{-j}(I)* 4^j))). - Peter Luschny, Apr 29 2013

a(n) ~ sqrt(2-sqrt(2)) * 2^(3*n+3/2) * n^(n+1/2) / (Pi^(n+1/2) * exp(n)). - Vaclav Kotesovec, Feb 25 2014

a(n) ~ GAMMA(n)*8^n/(Pi^n*(2*sqrt(4+2*sqrt(2)))). - Simon Plouffe, Nov 29 2018

Maple

per4 := proc(n) local j; 2*((1-I)/(1+I))^n*(1+add(binomial(n, j)* polylog(-j, I)*4^j, j=0..n)) end: A006873 := n -> Re(per4(n));
seq(A006873(i), i=0..11); # Peter Luschny, Apr 29 2013

Mathematica

mx = 17; Range[0, mx]! CoefficientList[ Series[ (Sin[x] + Cos[3x])/Cos[4x], {x, 0, mx}], x] (* Robert G. Wilson v, Apr 28 2013 *)

Prog

(PARI) x='x+O('x^66); Vec(serlaplace((sin(x)+cos(3*x))/cos(4*x))) \\ Joerg Arndt, Apr 28 2013
(Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Sin(x)+Cos(3*x))/Cos(4*x))); [Factorial(n-1)*b[n]: n in [1..m-1]]; // G. C. Greubel, Nov 29 2018
(Sage)
f=(sin(x) + cos(3*x))/cos(4*x)
g=f.taylor(x, 0, 50)
L=g.coefficients()
coeffs={c[1]:c[0]*factorial(c[1]) for c in L}
coeffs # G. C. Greubel, Nov 29 2018

Crossrefs

Cf. A007286, A007289, A225109.

Sequence in context: A173772 A178002 A288722 * A228695 A368295 A268063

Adjacent sequences: A006870 A006871 A006872 * A006874 A006875 A006876

Keyword

nonn

Author

N. J. A. Sloane and Simon Plouffe

Extensions

Added more terms, Joerg Arndt, Apr 28 2013

Status

approved