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Oscillating orbitals over n sectors (nonpositive values indicating there exist none).
19

%I #25 Jun 18 2022 08:40:48

%S -1,-1,0,0,2,10,10,70,42,378,168,1848,660,8580,2574,38610,10010,

%T 170170,38896,739024,151164,3174444,587860,13520780,2288132,57203300,

%U 8914800,240699600,34767720,1008263880,135727830,4207562730,530365050,17502046650,2074316640

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

%C 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 Peter Luschny, <a href="/A232500/a232500.pdf">Illustrating swinging orbitals</a>.

%F 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 E.g.f.: (1+x)*BesselI(0, 2*x)-2*(1+1/x)*BesselI(1, 2*x).

%F a(n) = (n!/k!^2)*(k-1)/(k+1) where k = floor(n/2).

%F 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 a(n) = A056040(n) * (1 - 2/(floor(n/2) + 1)).

%F a(n) = A056040(n) - 2*A057977(n).

%F 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 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 f := (z/(1-4*z^2)-3-1/z+1/z^2)/sqrt(1-4*z^2)-1/z^2+1/z;

%p seq(coeff(series(f, z, n+2), z, n), n=0..19);

%p g := (1+x)*BesselI(0, 2*x)-2*(1+1/x)*BesselI(1, 2*x);

%p seq(n!*coeff(series(g,x,n+2),x,n), n=0..19);

%t 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 *)

%o (Sage)

%o def A232500():

%o r, n = 1, 0

%o while True:

%o yield r*(n//2-1)/(n//2+1)

%o n += 1

%o r *= 4/n if is_even(n) else n

%o a = A232500(); [next(a) for i in range(36)]

%o (PARI) a(n) = n!/(n\2)!^2*(n\2-1)/(n\2+1) \\ _Charles R Greathouse IV_, Jul 30 2016

%Y Cf. A056040, A057977.

%K sign,nice

%O 0,5

%A _Peter Luschny_, Jan 05 2014