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%I #36 Mar 23 2020 12:09:34
%S 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,
%T 180,63,9,42,90,75,35,9,1,462,990,825,385,99,11,132,297,275,154,54,11,
%U 1,1716,3861,3575,2002,702,143,13,429,1001,1001,637,273,77,13,1
%N A statistic on orbital systems over n sectors: the number of orbitals which rise to maximum height k over the central circle.
%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 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.
%C T(n, k) is a subtriangle of the extended Catalan triangle A189231.
%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/TheLostCatalanNumbers">The lost Catalan numbers</a>
%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/Orbitals">Orbitals</a>
%e Triangle read by rows, n>=0. The length of row n is floor((n+2)/2).
%e [ n] [k=0,1,2,...] [row sum]
%e [ 0] [ 1] 1
%e [ 1] [ 1] 1
%e [ 2] [ 1, 1] 2
%e [ 3] [ 3, 3] 6
%e [ 4] [ 2, 3, 1] 6
%e [ 5] [ 10, 15, 5] 30
%e [ 6] [ 5, 9, 5, 1] 20
%e [ 7] [ 35, 63, 35, 7] 140
%e [ 8] [ 14, 28, 20, 7, 1] 70
%e [ 9] [126, 252, 180, 63, 9] 630
%e [10] [ 42, 90, 75, 35, 9, 1] 252
%e [11] [462, 990, 825, 385, 99, 11] 2772
%e [12] [132, 297, 275, 154, 54, 11, 1] 924
%e 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.
%p S := proc(n,k) option remember; `if`(k>n or k<0, 0, `if`(n=k, 1, S(n-1,k-1)+
%p modp(n-k,2)*S(n-1,k)+S(n-1,k+1))) end: T := (n,k) -> S(n,2*k);
%p seq(print(seq(T(n,k), k=0..iquo(n,2))), n=0..12);
%o (Sage)
%o from itertools import accumulate
%o # Brute force counting
%o def unit_orbitals(n):
%o sym_range = [i for i in range(-n+1, n, 2)]
%o for c in Combinations(sym_range, n):
%o P = Permutations([sgn(v) for v in c])
%o for p in P: yield p
%o def max_orbitals(n):
%o if n == 0: return [1]
%o S = [0]*((n+2)//2)
%o for u in unit_orbitals(n):
%o L = list(accumulate(u))
%o S[max(L)] += 1
%o return S
%o for n in (0..10): print(max_orbitals(n))
%Y Cf. A008313, A039599 (even rows), A047072, A056040 (row sums), A057977 (col 0), A063549 (col 0), A112467, A120730, A189230 (odd rows aerated), A189231, A232500.
%Y Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274708 (number of peaks), A274710 (number of turns), A274878 (span), A274879 (returns), A274880 (restarts), A274881 (ascent).
%K nonn,tabf
%O 0,5
%A _Peter Luschny_, Jul 09 2016
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