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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

Table of n, a(n) for n=0..64.

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

[ 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

[10] [0, 84, 96, 54, 16, 2] 252

[11] [0, 924, 1056, 594, 176,  22] 2772

[12] [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[1] > 0 if orb[0] == 0 else orb[0] > 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 [1]

    if n == 1: return [0, 1]

    S = [0]*(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.)