cp's OEIS Frontend

A232500 Oscillating orbitals over n sectors (nonpositive values indicating there exist none).

%I A232500 #27 Dec 16 2024 12:47:34
%S A232500 -1,-1,0,0,2,10,10,70,42,378,168,1848,660,8580,2574,38610,10010,
%T A232500 170170,38896,739024,151164,3174444,587860,13520780,2288132,57203300,
%U A232500 8914800,240699600,34767720,1008263880,135727830,4207562730,530365050,17502046650,2074316640
%N A232500 Oscillating orbitals over n sectors (nonpositive values indicating there exist none).
%C A232500 A planar orbital system is a family of concentric circles in a plane divided into n sectors. An orbital is a closed path consisting of arcs on these circles such that at each boundary of a sector the path jumps to the next inner or outer circle. One exception is allowed: if n is odd the path might continue on the same circle, but just once. After fixing one circle as the central circle there are three types of orbitals: a high orbital is always above the central circle, a low orbital is always below the central circle, and an oscillating orbital which is neither a high nor a low orbital. The number of all orbitals is A056040(n), the number of high orbitals, which is the same as the number of low orbitals, is A057977(n), and the number of oscillating orbitals is this a(n) (for n >= 4).
%H A232500 Peter Luschny, <a href="/A232500/a232500.pdf">Illustrating swinging orbitals</a>.
%F A232500 O.g.f.: (z/(1-4*z^2) - 3 - 1/z + 1/z^2)/sqrt(1-4*z^2) - 1/z^2 + 1/z.
%F A232500 E.g.f.: (1+x)*BesselI(0, 2*x)-2*(1+1/x)*BesselI(1, 2*x).
%F A232500 a(n) = (n!/k!^2)*(k-1)/(k+1) where k = floor(n/2).
%F A232500 Recurrence: If n > 4 then a(n) = a(n-1)*n if n is odd else a(n-1)*4*(n-2)/((n-4)*(n+2)).
%F A232500 a(n) = A056040(n) * (1 - 2/(floor(n/2) + 1)).
%F A232500 a(n) = A056040(n) - 2*A057977(n).
%F A232500 Asymptotic: log(a(n)) ~ (n*log(4) - log(Pi) - (-1)^n*(log(n/2) + 1/(2*n)))/2 + log(1 - 8/(2*n + 3 + (-1)^n)) for n >= 4.
%F A232500 D-finite with recurrence: +(n+2)*a(n) -n*a(n-1) +(-11*n+2)*a(n-2) +(9*n-16)*a(n-3) +20*(2*n-5)*a(n-4) +20*(-n+3)*a(n-5) +48*(-n+5)*a(n-6)=0. - _R. J. Mathar_, Feb 21 2020
%p A232500 f := (z/(1-4*z^2)-3-1/z+1/z^2)/sqrt(1-4*z^2)-1/z^2+1/z;
%p A232500 seq(coeff(series(f, z, n+2), z, n), n=0..19);
%p A232500 g := (1+x)*BesselI(0, 2*x)-2*(1+1/x)*BesselI(1, 2*x);
%p A232500 seq(n!*coeff(series(g,x,n+2),x,n), n=0..19);
%t A232500 sf[n_] := n!/Quotient[n, 2]!^2; a[n_] := sf[n]*(1-2/(Quotient[n, 2]+1)); Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Feb 11 2015 *)
%t A232500 nxt[{n_,a_}]:={n+1,If[EvenQ[n],a(n+1),a (4(n-1))/((n-3)(n+3))]}; Join[{-1,-1,0,0,2},NestList[nxt,{5,10},40][[;;,2]]] (* _Harvey P. Dale_, Dec 16 2024 *)
%o A232500 (Sage)
%o A232500 def A232500():
%o A232500     r, n = 1, 0
%o A232500     while True:
%o A232500         yield r*(n//2-1)/(n//2+1)
%o A232500         n += 1
%o A232500         r *= 4/n if is_even(n) else n
%o A232500 a = A232500(); [next(a) for i in range(36)]
%o A232500 (PARI) a(n) = n!/(n\2)!^2*(n\2-1)/(n\2+1) \\ _Charles R Greathouse IV_, Jul 30 2016
%Y A232500 Cf. A056040, A057977.
%K A232500 sign,nice
%O A232500 0,5
%A A232500 _Peter Luschny_, Jan 05 2014