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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.
10
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).
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
# Brute force counting, function unit_orbitals defined in A274709.
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
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, Jul 10 2016
STATUS
approved