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A274706
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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.
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10
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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
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OFFSET
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0,4
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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.
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PROG
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# 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)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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