A006873 Number of alternating 4-signed permutations.

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

Offset: 0

Author

N. J. A. Sloane and Simon Plouffe

Keywords

• nonn

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.

Crossrefs

Cf. A007286, A007289, A225109.

Programs

Magma

m:=50; R:=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

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 *)

PARI

x='x+O('x^66); Vec(serlaplace((sin(x)+cos(3*x))/cos(4*x))) \\ Joerg Arndt, Apr 28 2013

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

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

Extensions

Added more terms, Joerg Arndt, Apr 28 2013