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%I #20 Mar 05 2020 07:05:17
%S 1,0,2,2,2,2,0,4,0,2,6,12,4,2,6,0,12,0,4,0,4,20,60,12,12,12,4,20,0,40,
%T 0,12,0,8,0,10,70,280,40,60,36,24,40,10,70,0,140,0,40,0,24,0,20,0,28,
%U 252,1260,140,280,120,120,120,60,140,28,252,0,504,0
%N Triangle read by rows, number of orbitals classified with respect to the first zero crossing, n>=1, 1<=k<=n.
%C For the combinatorial definitions see A232500. An orbital w over n sectors has its first zero crossing at k if k is the smallest j such that the partial sum(1<=i<=j, w(i))) = 0, where w(i) are the jumps of the orbital represented by -1, 0, 1.
%F If n is even and k is odd then T(n, k) = 0 else if k = 1 then T(n, 1) = A056040(n-1) else T(n, k) = 2*A057977(k-2)*A056040(n-k).
%F T(n, n) = A241543(n).
%F T(n+1, 1) = A126869(n).
%F T(2*n, 2*n) = |A002420(n)|.
%F T(2*n+1, 1) = A000984(n).
%F T(2*n+1, n+1) = A241530(n).
%F T(2*n+2, 2) = A028329(n).
%F T(4*n, 2*n) = |A010370(n)|.
%F T(4*n, 4*n) = |A024491(n)|.
%F T(4*n+1, 1) = A001448(n).
%F T(4*n+1, 2*n+1) = A002894(n).
%e [1], [ 1]
%e [2], [ 0, 2]
%e [3], [ 2, 2, 2]
%e [4], [ 0, 4, 0, 2]
%e [5], [ 6, 12, 4, 2, 6]
%e [6], [ 0, 12, 0, 4, 0, 4]
%e [7], [20, 60, 12, 12, 12, 4, 20]
%p A241477 := proc(n, k)
%p if n = 0 then 1
%p elif k = 0 then 0
%p elif irem(n, 2) = 0 and irem(k, 2) = 1 then 0
%p elif k = 1 then (n-1)!/iquo(n-1,2)!^2
%p else 2*(n-k)!*(k-2)!/iquo(k,2)/(iquo(k-2,2)!*iquo(n-k,2)!)^2
%p fi end:
%p for n from 1 to 9 do seq(A241477(n, k), k=1..n) od;
%t T[n_, k_] := Which[n == 0, 1, k == 0, 0, Mod[n, 2] == 0 && Mod[k, 2] == 1, 0, k == 1, (n-1)!/Quotient[n-1, 2]!^2, True, 2*(n-k)!*(k-2)!/Quotient[k, 2]/(Quotient[k-2, 2]!*Quotient[n-k, 2]!)^2];
%t Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 20 2018, from Maple *)
%o (Sage)
%o def A241477_row(n):
%o if n == 0: return [1]
%o Z = [0]*n; T = [0] if is_odd(n) else []
%o for i in (1..n//2): T.append(-1); T.append(1)
%o for p in Permutations(T):
%o i = 0; s = p[0]
%o while s != 0: i += 1; s += p[i];
%o Z[i] += 1
%o return Z
%o for n in (1..9): A241477_row(n)
%Y Row sums: A056040.
%Y Cf. A232500.
%K nonn,tabl
%O 1,3
%A _Peter Luschny_, Apr 23 2014
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