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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 (list; table; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=1..69.

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-Fran├žois 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: A214666 A320471 A127444 * A268243 A159782 A268242

Adjacent sequences:  A241474 A241475 A241476 * A241478 A241479 A241480

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Apr 23 2014

STATUS

approved

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Last modified June 15 17:43 EDT 2019. Contains 324142 sequences. (Running on oeis4.)