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 A275325 Triangle read by rows: number of orbitals over n sectors which have a Catalan decomposition into k parts. 1

%I

%S 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,

%T 252,108,18,0,84,96,54,16,2,0,924,1056,594,176,22,0,264,330,220,88,20,

%U 2,0,3432,4290,2860,1144,260,26,0,858,1144,858,416,130,24,2

%N Triangle read by rows: number of orbitals over n sectors which have a Catalan decomposition into k parts.

%C The definition of an orbital system is given in A232500.

%C 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.

%C The number of orbitals over n sectors is counted by the swinging factorial A056040.

%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/Orbitals">Orbitals</a>

%F T(n,1) = 2*floor((n+2)/2)*n!/floor((n+2)/2)!^2 = A241543(n+2) for n>=2.

%F For odd n>1 T(n,1) = Sum_{k>=0} T(n+1,k).

%F A056040(n) - T(n,1) = A232500(n) for n>=2.

%F Main diagonal: T(n, floor(n/2)) = A266722(n) for n>1.

%F A275326(n,k) = ceiling(T(n,k)/2).

%e Table starts:

%e [ n] [k=0,1,2,...] [row sum]

%e [ 0]  1

%e [ 1] [0, 1] 1

%e [ 2] [0, 2] 2

%e [ 3] [0, 6] 6

%e [ 4] [0, 4, 2] 6

%e [ 5] [0, 20, 10] 30

%e [ 6] [0, 10, 8, 2] 20

%e [ 7] [0, 70, 56, 14] 140

%e [ 8] [0, 28, 28, 12, 2] 70

%e [ 9] [0, 252, 252, 108, 18] 630

%e  [0, 84, 96, 54, 16, 2] 252

%e  [0, 924, 1056, 594, 176, 22] 2772

%e  [0, 264, 330, 220, 88, 20, 2] 924

%e For example T(2*n, n) = 2 counts the Catalan decompositions

%e [[-1, 1], [1, -1], [-1, 1], ..., [(-1)^n, (-1)^(n+1)]] and

%e [[1, -1], [-1, 1], [1, -1], ..., [(-1)^(n+1), (-1)^n]].

%o (Sage)

%o # Brute force counting, function unit_orbitals defined in A274709.

%o def catalan_factors(P):

%o def bisect(orb):

%o i = 1

%o A = list(accumulate(orb))

%o if orb > 0 if orb == 0 else orb > 0:

%o while i < len(A) and A[i] >= 0: i += 1

%o else:

%o while i < len(A) and A[i] <= 0: i += 1

%o return i

%o R = []

%o while P <> []:

%o i = bisect(P)

%o R.append(P[:i])

%o P = P[i:]

%o return R

%o def orbital_factors(n):

%o if n == 0: return 

%o if n == 1: return [0, 1]

%o S = *(n//2 + 1)

%o for o in unit_orbitals(n):

%o S[len(catalan_factors(o))] += 1

%o return S

%o for n in (0..9): print orbital_factors(n)

%Y Cf. A056040 (row sum), A241543, A232500, A266722, A275326.

%Y 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).

%K nonn,tabf

%O 0,5

%A _Peter Luschny_, Aug 15 2016

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Last modified July 18 19:00 EDT 2019. Contains 325144 sequences. (Running on oeis4.)