This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A007289 M1873 #80 Jun 25 2025 08:46:07
%S A007289 1,2,8,46,352,3362,38528,515086,7869952,135274562,2583554048,
%T A007289 54276473326,1243925143552,30884386347362,825787662368768,
%U A007289 23657073914466766,722906928498737152,23471059057478981762,806875574817679474688,29279357851856595135406
%N A007289 Expansion of e.g.f. (sin(2*x) + cos(x)) / cos(3*x).
%C A007289 Arises in the enumeration of alternating 3-signed permutations.
%D A007289 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007289 Matthew House, <a href="/A007289/b007289.txt">Table of n, a(n) for n = 0..406</a> (terms 0..200 from Vincenzo Librandi)
%H A007289 R. Ehrenborg and M. A. Readdy, <a href="/A006873/a006873.pdf">Sheffer posets and r-signed permutations</a>, Preprint, 1994. (Annotated scanned copy)
%H A007289 Richard Ehrenborg and Margaret A. Readdy, <a href="http://www.labmath.uqam.ca/~annales/volumes/19-2.html">Sheffer posets and r-signed permutations</a>, Annales des Sciences Mathématiques du Québec, 19 (1995), 173-196.
%F A007289 E.g.f.: (sin(2*x) + cos(x)) / cos(3*x).
%F A007289 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
%F A007289 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
%F A007289 a(n) ~ n! * 2^(n+1)*3^(n+1/2)/Pi^(n+1). - _Vaclav Kotesovec_, Jun 15 2013
%F A007289 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
%F A007289 From _Seiichi Manyama_, Jun 25 2025: (Start)
%F A007289 E.g.f.: 1/(1 - 2 * sin(x)).
%F A007289 a(n) = Sum_{k=0..n} 2^k * k! * i^(n-k) * A136630(n,k), where i is the imaginary unit. (End)
%p A007289 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
%t A007289 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 *)
%o A007289 (PARI) my(x='x+O('x^66)); Vec(serlaplace((sin(2*x) + cos(x)) / cos(3*x))) \\ _Joerg Arndt_, Apr 28 2013
%o A007289 (Sage)
%o A007289 from mpmath import mp, polylog, im
%o A007289 mp.dps = 32; mp.pretty = True
%o A007289 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)))
%o A007289 def A007289(n) : return im(aperm3(n))
%o A007289 [int(A007289(n)) for n in (0..17)] # _Peter Luschny_, Apr 28 2013
%Y A007289 Row 3 of A349271.
%Y A007289 Cf. A006873, A007286, A225109, A000191 (bisection), A000436 (bisection).
%Y A007289 Cf. A000111, A136630, A352638.
%K A007289 nonn
%O A007289 0,2
%A A007289 _N. J. A. Sloane_ and _Simon Plouffe_