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A274709
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A statistic on orbital systems over n sectors: the number of orbitals which rise to maximum height k over the central circle.
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12
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1, 1, 1, 1, 3, 3, 2, 3, 1, 10, 15, 5, 5, 9, 5, 1, 35, 63, 35, 7, 14, 28, 20, 7, 1, 126, 252, 180, 63, 9, 42, 90, 75, 35, 9, 1, 462, 990, 825, 385, 99, 11, 132, 297, 275, 154, 54, 11, 1, 1716, 3861, 3575, 2002, 702, 143, 13, 429, 1001, 1001, 637, 273, 77, 13, 1
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OFFSET
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0,5
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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);
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PROG
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(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))
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CROSSREFS
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Cf. A008313, A039599 (even rows), A047072, A056040 (row sums), A057977 (col 0), A063549 (col 0), A112467, A120730, A189230 (odd rows aerated), A189231, A232500.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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