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A241477
Triangle read by rows, number of orbitals classified with respect to the first zero crossing, n>=1, 1<=k<=n.
13
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, 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, 252, 1260, 140, 280, 120, 120, 120, 60, 140, 28, 252, 0, 504, 0
OFFSET
1,3
COMMENTS
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.
FORMULA
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).
T(n, n) = A241543(n).
T(n+1, 1) = A126869(n).
T(2*n, 2*n) = |A002420(n)|.
T(2*n+1, 1) = A000984(n).
T(2*n+1, n+1) = A241530(n).
T(2*n+2, 2) = A028329(n).
T(4*n, 2*n) = |A010370(n)|.
T(4*n, 4*n) = |A024491(n)|.
T(4*n+1, 1) = A001448(n).
T(4*n+1, 2*n+1) = A002894(n).
EXAMPLE
[1], [ 1]
[2], [ 0, 2]
[3], [ 2, 2, 2]
[4], [ 0, 4, 0, 2]
[5], [ 6, 12, 4, 2, 6]
[6], [ 0, 12, 0, 4, 0, 4]
[7], [20, 60, 12, 12, 12, 4, 20]
MAPLE
A241477 := proc(n, k)
if n = 0 then 1
elif k = 0 then 0
elif irem(n, 2) = 0 and irem(k, 2) = 1 then 0
elif k = 1 then (n-1)!/iquo(n-1, 2)!^2
else 2*(n-k)!*(k-2)!/iquo(k, 2)/(iquo(k-2, 2)!*iquo(n-k, 2)!)^2
fi end:
for n from 1 to 9 do seq(A241477(n, k), k=1..n) od;
MATHEMATICA
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];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 20 2018, from Maple *)
PROG
(Sage)
def A241477_row(n):
if n == 0: return [1]
Z = [0]*n; T = [0] if is_odd(n) else []
for i in (1..n//2): T.append(-1); T.append(1)
for p in Permutations(T):
i = 0; s = p[0]
while s != 0: i += 1; s += p[i];
Z[i] += 1
return Z
for n in (1..9): A241477_row(n)
CROSSREFS
Row sums: A056040.
Cf. A232500.
Sequence in context: A320471 A333180 A127444 * A362485 A268243 A159782
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Apr 23 2014
STATUS
approved