OFFSET
0,5
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.
Note that (sum row_n) / row_n(0) = 1,1,2,2,3,3,4,4,..., i.e. the swinging factorials are multiples of the extended Catalan numbers A057977 generalizing the fact that the central binomials are multiples of the Catalan numbers.
T(n, k) is a subtriangle of the extended Catalan triangle A189231.
LINKS
EXAMPLE
Triangle read by rows, n>=0. The length of row n is floor((n+2)/2).
[ n] [k=0,1,2,...] [row sum]
[ 0] [ 1] 1
[ 1] [ 1] 1
[ 2] [ 1, 1] 2
[ 3] [ 3, 3] 6
[ 4] [ 2, 3, 1] 6
[ 5] [ 10, 15, 5] 30
[ 6] [ 5, 9, 5, 1] 20
[ 7] [ 35, 63, 35, 7] 140
[ 8] [ 14, 28, 20, 7, 1] 70
[ 9] [126, 252, 180, 63, 9] 630
[10] [ 42, 90, 75, 35, 9, 1] 252
[11] [462, 990, 825, 385, 99, 11] 2772
[12] [132, 297, 275, 154, 54, 11, 1] 924
T(6, 2) = 5 because the five orbitals [-1, 1, 1, 1, -1, -1], [1, -1, 1, 1, -1, -1], [1, 1, -1, -1, -1, 1], [1, 1, -1, -1, 1, -1], [1, 1, -1, 1, -1, -1] raise to maximal height of 2 over the central circle.
MAPLE
S := proc(n, k) option remember; `if`(k>n or k<0, 0, `if`(n=k, 1, S(n-1, k-1)+
modp(n-k, 2)*S(n-1, k)+S(n-1, k+1))) end: T := (n, k) -> S(n, 2*k);
seq(print(seq(T(n, k), k=0..iquo(n, 2))), n=0..12);
PROG
(Sage)
from itertools import accumulate
# Brute force counting
def unit_orbitals(n):
sym_range = [i for i in range(-n+1, n, 2)]
for c in Combinations(sym_range, n):
P = Permutations([sgn(v) for v in c])
for p in P: yield p
def max_orbitals(n):
if n == 0: return [1]
S = [0]*((n+2)//2)
for u in unit_orbitals(n):
L = list(accumulate(u))
S[max(L)] += 1
return S
for n in (0..10): print(max_orbitals(n))
CROSSREFS
Cf. A008313, A039599 (even rows), A047072, A056040 (row sums), A057977 (col 0), A063549 (col 0), A112467, A120730, A189230 (odd rows aerated), A189231, A232500.
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, Jul 09 2016
STATUS
approved