A344397 a(n) = Stirling2(n, floor(n/2)) * floor(n/2)!.

1, 0, 1, 1, 14, 30, 540, 1806, 40824, 186480, 5103000, 29607600, 953029440, 6711344640, 248619571200, 2060056318320, 86355926616960, 823172919528960, 38528927611574400, 415357755774998400, 21473732319740064000, 258323865658578720000, 14620825330739032204800

Offset

0,5

Links

Table of n, a(n) for n=0..22.

Formula

a(n) = [x^(floor(n/2)] F(n, x), the middle coefficient of the Fubini polynomial.

a(n) = Sum_{k=0..n/2} (-1)^k*binomial((2*n - 1)/4 + (-1)^n/4, k)*((2*n - 1)/4 + (-1)^n/4 - k)^n.

Maple

a := n -> add((-1)^k*binomial((2*n-1)/4 + (-1)^n/4, k)*((2*n-1)/4 + (-1)^n/4 - k)^n, k = 0..n/2):
# Alternative, via Fubini recurrence:
F := proc(n) option remember; if n = 0 then return 1 fi;
expand(add(binomial(n, k)*F(n - k)*x, k = 1..n)) end:
a := n -> coeff(F(n), x, iquo(n, 2));
seq(a(n), n = 0..22);

Mathematica

a[n_] := StirlingS2[n, Floor[n/2]] * Floor[n/2]!; Array[a, 23, 0] (* Amiram Eldar, May 22 2021 *)

Prog

(SageMath)
def a(n): return stirling_number2(n, n//2) * factorial(n//2)
print([a(n) for n in range(23)])

Crossrefs

Cf. A131689, A210029, A019538, A090582.

Sequence in context: A293391 A015222 A054103 * A161454 A367950 A156203

Adjacent sequences: A344394 A344395 A344396 * A344398 A344399 A344400

Keyword

nonn

Author

Peter Luschny, May 21 2021

Status

approved