A118920 -id:A118920 - cp's OEIS frontend

A274880 A statistic on orbital systems over n sectors: the number of orbitals with k restarts.

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

Offset: 0

Peter Luschny, Jul 11 2016

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.

			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] [1]                              1
[ 2] [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
[10] [84, 96, 54, 16, 2]            252
[11] [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].

Peter Luschny, Orbitals

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).

# uses[unit_orbitals from A274709]
from itertools import accumulate
# Brute force counting
def orbital_restart(n):
    if n == 0: return [1]
    S = [0]*((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))

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).

A118919 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that cross downwards the x-axis k times. (A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1)).

Original entry on oeis.org

1, 2, 5, 1, 14, 6, 42, 27, 1, 132, 110, 10, 429, 429, 65, 1, 1430, 1638, 350, 14, 4862, 6188, 1700, 119, 1, 16796, 23256, 7752, 798, 18, 58786, 87210, 33915, 4655, 189, 1, 208012, 326876, 144210, 24794, 1518, 22, 742900, 1225785, 600875, 123970, 10350

Offset: 0

Emeric Deutsch, May 06 2006

Row n contains 1+floor(n/2) terms. Row sums are the central binomial coefficients (A000984). T(n,0)=A000108(n+1) (the Catalan numbers). T(n,1)=A003517(n). T(n,2)=A003519(n). Sum(k*T(n,k),k>=0)=A008549(n-1). For both downward and upward crossings, see A118920.
Eigenvector is defined by: A119243(n) = Sum_{k=0..[n\2]} T(n,k)*A119243(k). This triangle is closely related to triangle A119245. - Paul D. Hanna, May 10 2006
Column k is the sum of columns 2k and 2k+1 of A039599. - Philippe Deléham, Nov 11 2008

			T(3,1)=6 because we have ud\dudu,ud\dduu,udud\du,uudd\du,ud\duud and duud\du (the downward crossings of the x-axis are shown by a back-slash \).
Triangle starts:
  1;
  2;
  5,1;
  14,6;
  42,27,1;
  132,110,10;

Cf. A000984, A000108, A003517, A003519, A008549, A118920.

Cf. A119243 (eigenvector), A119245 (variant).

T:=(n,k)->(2*k+1)*binomial(2*n+2,n-2*k)/(n+1): for n from 0 to 13 do seq(T(n,k),k=0..floor(n/2)) od; # yields sequence in triangular form

T(n,k)=if(n<2*k || k<0,0,(2*k+1)*binomial(2*n+2,n-2*k)/(n+1)) \\ Paul D. Hanna, May 10 2006

T(n,k)=(2k+1)binomial(2n+2,n-2k)/(n+1). G.f.=G(t,z)=C^2/(1-tz^2*C^4), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.

T(n,k)=A039599(n,2k)+A039599(n,2k+1). - Philippe Deléham, Nov 11 2008

cp's OEIS Frontend

A274880 A statistic on orbital systems over n sectors: the number of orbitals with k restarts.

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