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

-1, -1, 0, 0, 2, 10, 10, 70, 42, 378, 168, 1848, 660, 8580, 2574, 38610, 10010, 170170, 38896, 739024, 151164, 3174444, 587860, 13520780, 2288132, 57203300, 8914800, 240699600, 34767720, 1008263880, 135727830, 4207562730, 530365050, 17502046650, 2074316640

Offset: 0

Author

Peter Luschny, Jan 05 2014

Keywords

• sign
• nice

Comments

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).

Links

• Peter Luschny, Illustrating swinging orbitals.

Crossrefs

Cf. A056040, A057977.

Programs

Maple

f := (z/(1-4*z^2)-3-1/z+1/z^2)/sqrt(1-4*z^2)-1/z^2+1/z;
seq(coeff(series(f, z, n+2), z, n), n=0..19);
g := (1+x)*BesselI(0, 2*x)-2*(1+1/x)*BesselI(1, 2*x);
seq(n!*coeff(series(g,x,n+2),x,n), n=0..19);

Mathematica

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 *)
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 *)

PARI

a(n) = n!/(n\2)!^2*(n\2-1)/(n\2+1) \\ Charles R Greathouse IV, Jul 30 2016

Sage

def A232500():
    r, n = 1, 0
    while True:
        yield r*(n//2-1)/(n//2+1)
        n += 1
        r *= 4/n if is_even(n) else n
a = A232500(); [next(a) for i in range(36)]

Formula

O.g.f.: (z/(1-4*z^2) - 3 - 1/z + 1/z^2)/sqrt(1-4*z^2) - 1/z^2 + 1/z.

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

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

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)).

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

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

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.

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