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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. 12
 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 (list; graph; refs; listen; history; text; internal format)
 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 Peter Luschny, The lost Catalan numbers Peter Luschny, Orbitals 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  [ 42,  90,  75,  35,  9,  1] 252  [462, 990, 825, 385, 99, 11] 2772  [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      S = *((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. 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). Sequence in context: A120992 A129979 A228483 * A260896 A237347 A075017 Adjacent sequences:  A274706 A274707 A274708 * A274710 A274711 A274712 KEYWORD nonn,tabf AUTHOR Peter Luschny, Jul 09 2016 STATUS approved

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Last modified September 21 19:57 EDT 2020. Contains 337273 sequences. (Running on oeis4.)