A274710 - OEIS

%I #26 Mar 27 2020 06:57:42

%S 1,1,0,2,0,0,6,0,2,2,2,0,0,6,12,12,0,2,4,8,4,2,0,0,6,24,52,40,18,0,2,

%T 6,18,18,18,6,2,0,0,6,36,120,180,180,84,24,0,2,8,32,48,72,48,32,8,2,0,

%U 0,6,48,216,480,744,672,432,144,30,0,2,10,50,100,200,200,200,100,50,10,2

%N A statistic on orbital systems over n sectors: the number of orbitals which make k turns.

%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 'turn' of an orbital w takes place where signum(w[i]) is not equal to signum(w[i+1]).

%C A152659 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*C(n/2-1,(k-1+mod(k-1,2))/2)*C(n/2-1,(k-1-mod(k-1,2))/2) for k=0..n-1 (from A152659).

%e Triangle read by rows, n>=0. The length of row n is n for n>=1.

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

%e [0] [1]                                    1

%e [1] [1]                                    1

%e [2] [0, 2]                                 2

%e [3] [0, 0, 6]                              6

%e [4] [0, 2, 2,  2]                          6

%e [5] [0, 0, 6, 12,  12]                    30

%e [6] [0, 2, 4,  8,   4,   2]               20

%e [7] [0, 0, 6, 24,  52,  40,  18]         140

%e [8] [0, 2, 6, 18,  18,  18,   6,  2]      70

%e [9] [0, 0, 6, 36, 120, 180, 180, 84, 24] 630

%e T(5,2) = 6 because the six orbitals [-1, -1, 0, 1, 1], [-1, -1, 1, 1, 0], [0, -1, -1, 1, 1], [0, 1, 1, -1, -1], [1, 1, -1, -1, 0], [1, 1, 0, -1, -1] make 2 turns.

%o (Sage) # uses[unit_orbitals from A274709]

%o # Brute force counting

%o def orbital_turns(n):

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

%o     S = [0]*(n)

%o     for u in unit_orbitals(n):

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

%o         S[L] += 1

%o     return S

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

%Y Cf. A056040 (row sum), A152659, A232500.

%Y Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274708 (number of peaks), A274709 (max. height), A274878 (span), A274879 (returns), A274880 (restarts), A274881 (ascent).

%K nonn,tabf

%O 0,4

%A _Peter Luschny_, Jul 10 2016