A343581 a(n) = binomial(n, floor(n/2))*FallingFactorial(n - 1, n - floor(n/2)).

1, 0, 2, 6, 36, 240, 1200, 12600, 58800, 846720, 3810240, 69854400, 307359360, 6849722880, 29682132480, 779155977600, 3339239904000, 100919250432000, 428906814336000, 14668613050291200, 61934143990118400, 2364758225077248000, 9931984545324441600, 418798681661180620800

Offset

0,3

Comments

Partially ordered sets on n elements that consist entirely of floor(n/2) chains (nonempty, linearly ordered subsets).

Links

Table of n, a(n) for n=0..23.

Formula

a(n) = Sum_{j=floor(n/2)..n} |Stirling1(n, j)|*Stirling2(j, floor(n/2)).

a(n) = binomial(n - 1, floor(n/2) - 1)*n!/floor(n/2)!) for n >= 1, a(0) = 1.

a(n) = A271703(n, floor(n/2)).

Maple

a := n -> `if`(n=0, 1, binomial(n - 1, iquo(n, 2) - 1)*n!/iquo(n, 2)!):
seq(a(n), n = 0..21);

Prog

(SageMath)
def a(n): return binomial(n, n - n//2)*falling_factorial(n - 1, n - n//2)
print([a(n) for n in range(22)])
(PARI) a(n) = sum(j=n\2, n, abs(stirling(n, j, 1))*stirling(j, n\2, 2)); \\ Michel Marcus, Apr 22 2021

Crossrefs

Cf. A187535, A271703.

Keyword

nonn

Author

Peter Luschny, Apr 21 2021

Status

approved