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

%I A343581 #11 Dec 23 2024 09:51:52
%S A343581 1,0,2,6,36,240,1200,12600,58800,846720,3810240,69854400,307359360,
%T A343581 6849722880,29682132480,779155977600,3339239904000,100919250432000,
%U A343581 428906814336000,14668613050291200,61934143990118400,2364758225077248000,9931984545324441600,418798681661180620800
%N A343581 a(n) = binomial(n, floor(n/2))*FallingFactorial(n - 1, n - floor(n/2)).
%C A343581 Partially ordered sets on n elements that consist entirely of floor(n/2) chains (nonempty, linearly ordered subsets).
%F A343581 a(n) = Sum_{j=floor(n/2)..n} |Stirling1(n, j)|*Stirling2(j, floor(n/2)).
%F A343581 a(n) = binomial(n - 1, floor(n/2) - 1)*n!/floor(n/2)! for n >= 1, a(0) = 1.
%F A343581 a(n) = A271703(n, floor(n/2)).
%p A343581 a := n -> `if`(n=0, 1, binomial(n - 1, iquo(n,2) - 1)*n!/iquo(n, 2)!):
%p A343581 seq(a(n), n = 0..21);
%o A343581 (SageMath)
%o A343581 def a(n): return binomial(n, n - n//2)*falling_factorial(n - 1, n - n//2)
%o A343581 print([a(n) for n in range(22)])
%o A343581 (PARI) a(n) = sum(j=n\2, n, abs(stirling(n, j, 1))*stirling(j, n\2, 2)); \\ _Michel Marcus_, Apr 22 2021
%Y A343581 Cf. A187535, A271703.
%K A343581 nonn
%O A343581 0,3
%A A343581 _Peter Luschny_, Apr 21 2021