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 A274880 A statistic on orbital systems over n sectors: the number of orbitals with k restarts. 10
 1, 1, 2, 5, 1, 4, 2, 18, 11, 1, 10, 8, 2, 65, 57, 17, 1, 28, 28, 12, 2, 238, 252, 116, 23, 1, 84, 96, 54, 16, 2, 882, 1050, 615, 195, 29, 1, 264, 330, 220, 88, 20, 2, 3300, 4257, 2915, 1210, 294, 35, 1, 858, 1144, 858, 416, 130, 24, 2, 12441, 17017, 13013, 6461, 2093, 413, 41, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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. A 'restart' of an orbital is a raise which starts from the central circle. A118920 is a subtriangle. LINKS Peter Luschny, Orbitals FORMULA For even n>0: T(n,k) = 4*(k+1)*binomial(n,n/2-k-1)/n for k=0..n/2-1 (from A118920). EXAMPLE Triangle read by rows, n>=0. The length of row n is floor((n+1)/2) for n>=1. [n] [k=0,1,2,...] [row sum] [ 0]  1 [ 1]  1 [ 2]  2 [ 3] [5, 1] 6 [ 4] [4, 2] 6 [ 5] [18, 11, 1] 30 [ 6] [10, 8, 2] 20 [ 7] [65, 57, 17, 1] 140 [ 8] [28, 28, 12, 2] 70 [ 9] [238, 252, 116, 23, 1] 630  [84, 96, 54, 16, 2] 252  [882, 1050, 615, 195, 29, 1] 2772 T(6, 2) = 2 because there are two orbitals over 6 segments which have 2 ascents: [-1, 1, 1, -1, 1, -1] and [1, -1, 1, -1, 1, -1]. PROG (Sage) # Brute force counting, function unit_orbitals defined in A274709. def orbital_restart(n):     if n == 0: return      S = *((n+1)//2)     for u in unit_orbitals(n):         A = list(accumulate(u))         L = [1 if A[i] == 0 and A[i+1] == 1  else 0 for i in (0..n-2)]         S[sum(L)] += 1     return S for n in (0..12): print orbital_restart(n) CROSSREFS Cf. A056040 (row sum), A118920, A232500. Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274708 (peaks), A274709 (max. height), A274710 (number of turns), A274878 (span), A274879 (returns), A274881 (ascent). Sequence in context: A316131 A153726 A229339 * A034005 A161688 A163331 Adjacent sequences:  A274877 A274878 A274879 * A274881 A274882 A274883 KEYWORD nonn,tabf AUTHOR Peter Luschny, Jul 11 2016 STATUS approved

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Last modified June 18 14:52 EDT 2019. Contains 324213 sequences. (Running on oeis4.)