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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.

%I A274706 #31 May 20 2025 21:08:57
%S A274706 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,
%T A274706 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,
%U A274706 48,64,48,48,40,44,32,36,24,22,16,16,8,10,8,4,4,2
%N 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.
%C A274706 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.
%C A274706 An orbital is balanced if its integral is 0 (A241810).
%H A274706 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/Orbitals">Orbitals</a>
%e A274706 The length of row n is 1+floor(n^2//4).
%e A274706 The triangle begins:
%e A274706   [n] [k=0,1,2,...] [row sum]
%e A274706   [0] [1] 1
%e A274706   [1] [1] 1
%e A274706   [2] [0, 2] 2
%e A274706   [3] [0, 4, 2] 6
%e A274706   [4] [2, 0, 2, 0, 2] 6
%e A274706   [5] [6, 4, 6, 4, 4, 4, 2] 30
%e A274706   [6] [0, 6, 0, 6, 0, 4, 0, 2, 0, 2] 20
%e A274706   [7] [6, 24, 16, 20, 14, 16, 12, 8, 6, 8, 4, 4, 2] 140
%e A274706   [8] [8, 0, 14, 0, 14, 0, 10, 0, 10, 0, 6, 0, 4, 0, 2, 0, 2] 70
%e A274706 T(5, 4) = 4 because the integral of four orbitals have the absolute value 4:
%e A274706   Integral([-1, -1, 1, 1, 0]) = -4, Integral([0, -1, -1, 1, 1]) = -4,
%e A274706   Integral([0, 1, 1, -1, -1]) = 4, Integral([1, 1, -1, -1, 0]) = 4.
%o A274706 (Sage)
%o A274706 from itertools import accumulate
%o A274706 # Brute force counting
%o A274706 def unit_orbitals(n):
%o A274706     sym_range = [i for i in range(-n+1, n, 2)]
%o A274706     for c in Combinations(sym_range, n):
%o A274706         P = Permutations([sgn(v) for v in c])
%o A274706         for p in P: yield p
%o A274706 def orbital_integral(n):
%o A274706     if n == 0: return [1]
%o A274706     S = [0]*(1+floor(n^2//4))
%o A274706     for u in unit_orbitals(n):
%o A274706         L = list(accumulate(accumulate(u)))
%o A274706         S[abs(L[-1])] += 1
%o A274706     return S
%o A274706 for n in (0..8): print(orbital_integral(n))
%Y A274706 Cf. A056040 (row sum), A232500, A241810 (col. 0), A242087.
%Y A274706 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).
%K A274706 nonn,tabf
%O A274706 0,4
%A A274706 _Peter Luschny_, Jul 10 2016