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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

Table of n, a(n) for n=0..63.

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

[10] [ 42,  90,  75,  35,  9,  1] 252

[11] [462, 990, 825, 385, 99, 11] 2772

[12] [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)

# 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 [1]

    S = [0]*((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 June 18 15:13 EDT 2019. Contains 324213 sequences. (Running on oeis4.)