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 A275325 Triangle read by rows: number of orbitals over n sectors which have a Catalan decomposition into k parts. 1
 1, 0, 1, 0, 2, 0, 6, 0, 4, 2, 0, 20, 10, 0, 10, 8, 2, 0, 70, 56, 14, 0, 28, 28, 12, 2, 0, 252, 252, 108, 18, 0, 84, 96, 54, 16, 2, 0, 924, 1056, 594, 176, 22, 0, 264, 330, 220, 88, 20, 2, 0, 3432, 4290, 2860, 1144, 260, 26, 0, 858, 1144, 858, 416, 130, 24, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS The definition of an orbital system is given in A232500. The Catalan decomposition of an orbital w is a list of orbitals which are alternately entirely above or below the main circle ('above' and 'below' in the weak sense) such that their concatenation equals w. If a zero is on the border of two orbitals then it is allocated to the first one. By convention T(0,0) = 1. The number of orbitals over n sectors is counted by the swinging factorial A056040. LINKS Peter Luschny, Orbitals FORMULA T(n,1) = 2*floor((n+2)/2)*n!/floor((n+2)/2)!^2 = A241543(n+2) for n>=2. For odd n>1 T(n,1) = Sum_{k>=0} T(n+1,k). A056040(n) - T(n,1) = A232500(n) for n>=2. Main diagonal: T(n, floor(n/2)) = A266722(n) for n>1. A275326(n,k) = ceiling(T(n,k)/2). EXAMPLE Table starts: [ n] [k=0,1,2,...] [row sum] [ 0]  1 [ 1] [0, 1] 1 [ 2] [0, 2] 2 [ 3] [0, 6] 6 [ 4] [0, 4, 2] 6 [ 5] [0, 20, 10] 30 [ 6] [0, 10, 8, 2] 20 [ 7] [0, 70, 56, 14] 140 [ 8] [0, 28, 28, 12, 2] 70 [ 9] [0, 252, 252, 108, 18] 630  [0, 84, 96, 54, 16, 2] 252  [0, 924, 1056, 594, 176,  22] 2772  [0, 264, 330, 220, 88, 20, 2] 924 For example T(2*n, n) = 2 counts the Catalan decompositions [[-1, 1], [1, -1], [-1, 1], ..., [(-1)^n, (-1)^(n+1)]] and [[1, -1], [-1, 1], [1, -1], ..., [(-1)^(n+1), (-1)^n]]. PROG (Sage) # Brute force counting, function unit_orbitals defined in A274709. def catalan_factors(P):     def bisect(orb):         i = 1         A = list(accumulate(orb))         if orb > 0 if orb == 0 else orb > 0:             while i < len(A) and A[i] >= 0: i += 1         else:             while i < len(A) and A[i] <= 0: i += 1         return i     R = []     while P <> []:         i = bisect(P)         R.append(P[:i])         P = P[i:]     return R def orbital_factors(n):     if n == 0: return      if n == 1: return [0, 1]     S = *(n//2 + 1)     for o in unit_orbitals(n):         S[len(catalan_factors(o))] += 1     return S for n in (0..9): print orbital_factors(n) CROSSREFS Cf. A056040 (row sum), A241543, A232500, A266722, A275326. Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274709 (max. height), A274710 (number of turns), A274878 (span), A274879 (returns), A274880 (restarts), A274881 (ascent). Sequence in context: A274881 A303638 A162974 * A300227 A290971 A178636 Adjacent sequences:  A275322 A275323 A275324 * A275326 A275327 A275328 KEYWORD nonn,tabf AUTHOR Peter Luschny, Aug 15 2016 STATUS approved

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Last modified June 17 19:10 EDT 2019. Contains 324198 sequences. (Running on oeis4.)