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

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, 84, 96, 54, 16, 2, 882, 1050, 615, 195, 29, 1, 264, 330, 220, 88, 20, 2, 3300, 4257, 2915, 1210, 294, 35, 1, 858, 1144, 858, 416, 130, 24, 2, 12441, 17017, 13013, 6461, 2093, 413, 41, 1

Offset: 0

Author

Peter Luschny, Jul 11 2016

Keywords

• nonn
• tabf

Comments

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.

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

A118920 is a subtriangle.

Examples

			Triangle read by rows, n>=0. The length of row n is floor((n+1)/2) for n>=1.
[n] [k=0,1,2,...]                 [row sum]
[ 0] [1]                              1
[ 1] [1]                              1
[ 2] [2]                              2
[ 3] [5, 1]                           6
[ 4] [4, 2]                           6
[ 5] [18, 11, 1]                     30
[ 6] [10, 8, 2]                      20
[ 7] [65, 57, 17, 1]                140
[ 8] [28, 28, 12, 2]                 70
[ 9] [238, 252, 116, 23, 1]         630
[10] [84, 96, 54, 16, 2]            252
[11] [882, 1050, 615, 195, 29, 1]  2772
T(6, 2) = 2 because there are two orbitals over 6 segments which have 2 ascents:
[-1, 1, 1, -1, 1, -1] and [1, -1, 1, -1, 1, -1].
		

Links

• Peter Luschny, Orbitals

Crossrefs

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

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).

Programs

Sage

# uses[unit_orbitals from A274709]
from itertools import accumulate
# Brute force counting
def orbital_restart(n):
    if n == 0: return [1]
    S = [0]*((n+1)//2)
    for u in unit_orbitals(n):
        A = list(accumulate(u))
        L = [1 if A[i] == 0 and A[i+1] == 1  else 0 for i in (0..n-2)]
        S[sum(L)] += 1
    return S
for n in (0..12): print(orbital_restart(n))

Formula

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).