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A274879 A statistic on orbital systems over n sectors: the number of orbitals with k returns. 10

%I

%S 1,1,2,2,4,2,4,6,12,12,4,8,8,20,40,48,32,10,20,24,16,70,140,180,160,

%T 80,28,56,72,64,32,252,504,672,672,480,192,84,168,224,224,160,64,924,

%U 1848,2520,2688,2240,1344,448,264,528,720,768,640,384,128

%N A statistic on orbital systems over n sectors: the number of orbitals with k returns.

%C The definition of an orbital system is given in A232500 (see also the illustration there). The number of orbitals over n sectors is counted by the swinging factorial A056040.

%C When a segment of an orbital starts at a point on the central circle this point is called a 'return' of the orbital if it is not the origin.

%C If an orbital touches the central circle only in the origin it is called a prime orbital. Column 0 counts the prime orbitals over n sectors.

%C A108747 is a subtriangle.

%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/Orbitals">Orbitals</a>

%F For even n>0: T(n,k) = 2^(k+1)*(k+1)*binomial(n-k-1,n/2)/(n-k-1) for k=0..n/2-1 (from A108747).

%e Triangle read by rows, n>=0. The length of row n is floor((n+1)/2) for n>=1.

%e [ n] [k=0,1,2,...] [row sum]

%e [ 0] [1] 1

%e [ 1] [1] 1

%e [ 2] [2] 2

%e [ 3] [2, 4] 6

%e [ 4] [2, 4] 6

%e [ 5] [6, 12, 12] 30

%e [ 6] [4, 8, 8] 20

%e [ 7] [20, 40, 48, 32] 140

%e [ 8] [10, 20, 24, 16] 70

%e [ 9] [70, 140, 180, 160, 80] 630

%e [10] [28, 56, 72, 64, 32] 252

%e [11] [252, 504, 672, 672, 480, 192] 2772

%e T(6,0) = 4 because the following 4 orbitals stay above or below the central

%e circle: [-1, -1, -1, 1, 1, 1], [-1, -1, 1, -1, 1, 1], [1, 1, -1, 1, -1, -1],

%e [1, 1, 1, -1, -1, -1].

%o (Sage)

%o # Brute force counting, function unit_orbitals defined in A274709.

%o def orbital_returns(n):

%o if n == 0: return [1]

%o S = [0]*((n+1)//2)

%o for u in unit_orbitals(n):

%o L = list(accumulate(u))

%o Z = len(filter(lambda z: z == 0, L))

%o S[Z-1] += 1 # exclude origin

%o return S

%o for n in (0..10): print orbital_returns(n)

%Y Cf. A056040 (row sum), A108747, A232500, A241543 (col. 0).

%Y Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274708 (peaks), A274709 (max. height), A274710 (number of turns), A274878 (span), A274880 (restarts), A274881 (ascent).

%K nonn,tabf

%O 0,3

%A _Peter Luschny_, Jul 11 2016

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Last modified November 15 11:18 EST 2019. Contains 329144 sequences. (Running on oeis4.)