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A274709 A statistic on orbital systems over n sectors: the number of orbitals which rise to maximum height k over the central circle.

%I A274709 #36 Mar 23 2020 12:09:34
%S A274709 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 A274709 180,63,9,42,90,75,35,9,1,462,990,825,385,99,11,132,297,275,154,54,11,
%U A274709 1,1716,3861,3575,2002,702,143,13,429,1001,1001,637,273,77,13,1
%N A274709 A statistic on orbital systems over n sectors: the number of orbitals which rise to maximum height k over the central circle.
%C A274709 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 A274709 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 A274709 T(n, k) is a subtriangle of the extended Catalan triangle A189231.
%H A274709 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/TheLostCatalanNumbers">The lost Catalan numbers</a>
%H A274709 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/Orbitals">Orbitals</a>
%e A274709 Triangle read by rows, n>=0. The length of row n is floor((n+2)/2).
%e A274709 [ n] [k=0,1,2,...] [row sum]
%e A274709 [ 0] [  1] 1
%e A274709 [ 1] [  1] 1
%e A274709 [ 2] [  1,   1] 2
%e A274709 [ 3] [  3,   3] 6
%e A274709 [ 4] [  2,   3,   1] 6
%e A274709 [ 5] [ 10,  15,   5] 30
%e A274709 [ 6] [  5,   9,   5,   1] 20
%e A274709 [ 7] [ 35,  63,  35,   7] 140
%e A274709 [ 8] [ 14,  28,  20,   7,  1] 70
%e A274709 [ 9] [126, 252, 180,  63,  9] 630
%e A274709 [10] [ 42,  90,  75,  35,  9,  1] 252
%e A274709 [11] [462, 990, 825, 385, 99, 11] 2772
%e A274709 [12] [132, 297, 275, 154, 54, 11, 1] 924
%e A274709 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 A274709 S := proc(n,k) option remember; `if`(k>n or k<0, 0, `if`(n=k, 1, S(n-1,k-1)+
%p A274709 modp(n-k,2)*S(n-1,k)+S(n-1,k+1))) end: T := (n,k) -> S(n,2*k);
%p A274709 seq(print(seq(T(n,k), k=0..iquo(n,2))), n=0..12);
%o A274709 (Sage)
%o A274709 from itertools import accumulate
%o A274709 # Brute force counting
%o A274709 def unit_orbitals(n):
%o A274709     sym_range = [i for i in range(-n+1, n, 2)]
%o A274709     for c in Combinations(sym_range, n):
%o A274709         P = Permutations([sgn(v) for v in c])
%o A274709         for p in P: yield p
%o A274709 def max_orbitals(n):
%o A274709     if n == 0: return [1]
%o A274709     S = [0]*((n+2)//2)
%o A274709     for u in unit_orbitals(n):
%o A274709         L = list(accumulate(u))
%o A274709         S[max(L)] += 1
%o A274709     return S
%o A274709 for n in (0..10): print(max_orbitals(n))
%Y A274709 Cf. A008313, A039599 (even rows), A047072, A056040 (row sums), A057977 (col 0), A063549 (col 0), A112467, A120730, A189230 (odd rows aerated), A189231, A232500.
%Y A274709 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 A274709 nonn,tabf
%O A274709 0,5
%A A274709 _Peter Luschny_, Jul 09 2016