cp's OEIS Frontend

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

%I A274880 #22 Mar 23 2020 17:33:32
%S A274880 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 A274880 84,96,54,16,2,882,1050,615,195,29,1,264,330,220,88,20,2,3300,4257,
%U A274880 2915,1210,294,35,1,858,1144,858,416,130,24,2,12441,17017,13013,6461,2093,413,41,1
%N A274880 A statistic on orbital systems over n sectors: the number of orbitals with k restarts.
%C A274880 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 A274880 A 'restart' of an orbital is a raise which starts from the central circle.
%C A274880 A118920 is a subtriangle.
%H A274880 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/Orbitals">Orbitals</a>
%F A274880 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 A274880 Triangle read by rows, n>=0. The length of row n is floor((n+1)/2) for n>=1.
%e A274880 [n] [k=0,1,2,...]                 [row sum]
%e A274880 [ 0] [1]                              1
%e A274880 [ 1] [1]                              1
%e A274880 [ 2] [2]                              2
%e A274880 [ 3] [5, 1]                           6
%e A274880 [ 4] [4, 2]                           6
%e A274880 [ 5] [18, 11, 1]                     30
%e A274880 [ 6] [10, 8, 2]                      20
%e A274880 [ 7] [65, 57, 17, 1]                140
%e A274880 [ 8] [28, 28, 12, 2]                 70
%e A274880 [ 9] [238, 252, 116, 23, 1]         630
%e A274880 [10] [84, 96, 54, 16, 2]            252
%e A274880 [11] [882, 1050, 615, 195, 29, 1]  2772
%e A274880 T(6, 2) = 2 because there are two orbitals over 6 segments which have 2 ascents:
%e A274880 [-1, 1, 1, -1, 1, -1] and [1, -1, 1, -1, 1, -1].
%o A274880 (Sage) # uses[unit_orbitals from A274709]
%o A274880 from itertools import accumulate
%o A274880 # Brute force counting
%o A274880 def orbital_restart(n):
%o A274880     if n == 0: return [1]
%o A274880     S = [0]*((n+1)//2)
%o A274880     for u in unit_orbitals(n):
%o A274880         A = list(accumulate(u))
%o A274880         L = [1 if A[i] == 0 and A[i+1] == 1  else 0 for i in (0..n-2)]
%o A274880         S[sum(L)] += 1
%o A274880     return S
%o A274880 for n in (0..12): print(orbital_restart(n))
%Y A274880 Cf. A056040 (row sum), A118920, A232500.
%Y A274880 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 A274880 nonn,tabf
%O A274880 0,3
%A A274880 _Peter Luschny_, Jul 11 2016