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

0,4

Comments

For the combinatorial definitions see A232500. The absolute integral of an orbital w over n sectors is abs(Sum_{1<=k<=n} Sum_{1<=i<=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).

Links

Table of n, a(n) for n=0..79.

Peter Luschny, Orbitals

Example

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.

Prog

(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))

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).

Sequence in context: A339941 A211318 A324239 * A037035 A159984 A240697

Adjacent sequences: A274703 A274704 A274705 * A274707 A274708 A274709

Keyword

nonn,tabf

Author

Peter Luschny, Jul 10 2016

Status

approved