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%I M4430 #53 Sep 08 2022 08:44:35
%S 1,1,7,47,497,6241,95767,1704527,34741217,796079041,20273087527,
%T 567864586607,17352768515537,574448847467041,20479521468959287,
%U 782259922208550287,31872138933891307457,1379749466246228538241,63243057486503656319047,3059895336952604166395567
%N Number of alternating 4-signed permutations.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A006873/b006873.txt">Table of n, a(n) for n = 0..200</a>
%H R. Ehrenborg and M. A. Readdy, <a href="/A006873/a006873.pdf">Sheffer posets and r-signed permutations</a>, Preprint, 1994. (Annotated scanned copy)
%H Richard Ehrenborg and Margaret A. Readdy, <a href="http://www.labmath.uqam.ca/~annales/volumes/19-2/PDF/173-196.pdf">Sheffer posets and r-signed permutations</a>, Annales des Sciences Mathématiques du Québec, 19 (1995), 173-196.
%F E.g.f.: (sin(x) + cos(3*x)) / cos(4*x). - _M. F. Hasler_, Apr 28 2013
%F 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
%F 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
%F a(n) ~ GAMMA(n)*8^n/(Pi^n*(2*sqrt(4+2*sqrt(2)))). - _Simon Plouffe_, Nov 29 2018
%p 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));
%p seq(A006873(i), i=0..11); # _Peter Luschny_, Apr 29 2013
%t mx = 17; Range[0, mx]! CoefficientList[ Series[ (Sin[x] + Cos[3x])/Cos[4x], {x, 0, mx}], x] (* _Robert G. Wilson v_, Apr 28 2013 *)
%o (PARI) x='x+O('x^66); Vec(serlaplace((sin(x)+cos(3*x))/cos(4*x))) \\ _Joerg Arndt_, Apr 28 2013
%o (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
%o (Sage)
%o f=(sin(x) + cos(3*x))/cos(4*x)
%o g=f.taylor(x,0,50)
%o L=g.coefficients()
%o coeffs={c[1]:c[0]*factorial(c[1]) for c in L}
%o coeffs # _G. C. Greubel_, Nov 29 2018
%Y Cf. A007286, A007289, A225109.
%K nonn
%O 0,3
%A _N. J. A. Sloane_ and _Simon Plouffe_
%E Added more terms, _Joerg Arndt_, Apr 28 2013
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