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A232500 Oscillating orbitals over n sectors (nonpositive values indicating there exist none). 19
-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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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

Table of n, a(n) for n=0..34.

Peter Luschny, Illustrating swinging orbitals

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!/[n/2]!^2*([n/2]-1)/([n/2]+1).

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/([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.

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-Fran├žois Alcover, Feb 11 2015 *)

PROG

(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(); [a.next() for i in range(36)]

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

CROSSREFS

Cf. A056040, A057977.

Sequence in context: A206486 A067046 A066394 * A033466 A193181 A222638

Adjacent sequences:  A232497 A232498 A232499 * A232501 A232502 A232503

KEYWORD

sign,nice

AUTHOR

Peter Luschny, Jan 05 2014

STATUS

approved

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Last modified June 26 10:45 EDT 2019. Contains 324375 sequences. (Running on oeis4.)