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A274880 A statistic on orbital systems over n sectors: the number of orbitals with k restarts. 10

%I

%S 1,1,2,5,1,4,2,18,11,1,10,8,2,65,57,17,1,28,28,12,2,238,252,116,23,1,

%T 84,96,54,16,2,882,1050,615,195,29,1,264,330,220,88,20,2,3300,4257,

%U 2915,1210,294,35,1,858,1144,858,416,130,24,2,12441,17017,13013,6461,2093,413,41,1

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

%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 A 'restart' of an orbital is a raise which starts from the central circle.

%C A118920 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) = 4*(k+1)*binomial(n,n/2-k-1)/n for k=0..n/2-1 (from A118920).

%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] [5, 1] 6

%e [ 4] [4, 2] 6

%e [ 5] [18, 11, 1] 30

%e [ 6] [10, 8, 2] 20

%e [ 7] [65, 57, 17, 1] 140

%e [ 8] [28, 28, 12, 2] 70

%e [ 9] [238, 252, 116, 23, 1] 630

%e [10] [84, 96, 54, 16, 2] 252

%e [11] [882, 1050, 615, 195, 29, 1] 2772

%e T(6, 2) = 2 because there are two orbitals over 6 segments which have 2 ascents:

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

%o (Sage)

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

%o def orbital_restart(n):

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

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

%o for u in unit_orbitals(n):

%o A = list(accumulate(u))

%o L = [1 if A[i] == 0 and A[i+1] == 1 else 0 for i in (0..n-2)]

%o S[sum(L)] += 1

%o return S

%o for n in (0..12): print orbital_restart(n)

%Y Cf. A056040 (row sum), A118920, A232500.

%Y Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274708 (peaks), A274709 (max. height), A274710 (number of turns), A274878 (span), A274879 (returns), A274881 (ascent).

%K nonn,tabf

%O 0,3

%A _Peter Luschny_, Jul 11 2016

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Last modified July 17 16:38 EDT 2019. Contains 325107 sequences. (Running on oeis4.)