A274706 Irregular triangle read by rows. T(n,k) (n >= 0) is a statistic on orbital systems over n sectors: the number of orbitals which have an integral whose absolute value is k.

1, 1, 0, 2, 0, 4, 2, 2, 0, 2, 0, 2, 6, 4, 6, 4, 4, 4, 2, 0, 6, 0, 6, 0, 4, 0, 2, 0, 2, 6, 24, 16, 20, 14, 16, 12, 8, 6, 8, 4, 4, 2, 8, 0, 14, 0, 14, 0, 10, 0, 10, 0, 6, 0, 4, 0, 2, 0, 2, 36, 52, 68, 48, 64, 48, 48, 40, 44, 32, 36, 24, 22, 16, 16, 8, 10, 8, 4, 4, 2

Offset: 0

Author

Peter Luschny, Jul 10 2016

Keywords

• nonn
• tabf

Comments

For the combinatorial definitions see A232500. The absolute integral of an orbital w over n sectors is abs(Sum_{k=1..n} Sum_{i=1..k} w(i)) where w(i) are the jumps of the orbital represented by -1, 0, 1.

An orbital is balanced if its integral is 0 (A241810).

Examples

			The length of row n is 1+floor(n^2//4).
The triangle begins:
  [n] [k=0,1,2,...] [row sum]
  [0] [1] 1
  [1] [1] 1
  [2] [0, 2] 2
  [3] [0, 4, 2] 6
  [4] [2, 0, 2, 0, 2] 6
  [5] [6, 4, 6, 4, 4, 4, 2] 30
  [6] [0, 6, 0, 6, 0, 4, 0, 2, 0, 2] 20
  [7] [6, 24, 16, 20, 14, 16, 12, 8, 6, 8, 4, 4, 2] 140
  [8] [8, 0, 14, 0, 14, 0, 10, 0, 10, 0, 6, 0, 4, 0, 2, 0, 2] 70
T(5, 4) = 4 because the integral of four orbitals have the absolute value 4:
  Integral([-1, -1, 1, 1, 0]) = -4, Integral([0, -1, -1, 1, 1]) = -4,
  Integral([0, 1, 1, -1, -1]) = 4, Integral([1, 1, -1, -1, 0]) = 4.
		

Links

• Peter Luschny, Orbitals

Crossrefs

Cf. A056040 (row sum), A232500, A241810 (col. 0), A242087.

Other orbital statistics: A241477 (first zero crossing), A274708 (number of peaks), A274709 (max. height), A274710 (number of turns), A274878 (span), A274879 (returns), A274880 (restarts), A274881 (ascent).

Programs

Sage

from itertools import accumulate
# 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 orbital_integral(n):
    if n == 0: return [1]
    S = [0]*(1+floor(n^2//4))
    for u in unit_orbitals(n):
        L = list(accumulate(accumulate(u)))
        S[abs(L[-1])] += 1
    return S
for n in (0..8): print(orbital_integral(n))