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a(n) = Product_{k=1..n} binomial(n, floor(n/k)).
3

%I #22 Feb 05 2024 03:10:12

%S 1,1,2,9,96,1250,64800,1764735,224788480,22499086176,6123600000000,

%T 408514437465750,1308805762115174400,133962125607455951520,

%U 99335199198879310098432,113040832521732593994140625,425230288403106927476736000000,72623663171934137824096600064000

%N a(n) = Product_{k=1..n} binomial(n, floor(n/k)).

%H G. C. Greubel, <a href="/A345466/b345466.txt">Table of n, a(n) for n = 0..160</a>

%F log(a(n)) ~ n * log(n)^2 / 2. - _Vaclav Kotesovec_, Jun 21 2021

%F a(n) = Product_{k=1..n} ((n+1)/k - 1)^floor(n/k). - _Vaclav Kotesovec_, Jun 24 2021

%t Table[Product[Binomial[n, Floor[n/k]], {k, 1, n}], {n, 0, 20}]

%t Table[Product[((n + 1)/k - 1)^Floor[n/k], {k, 1, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jun 24 2021 *)

%o (Magma) [n eq 0 select 1 else (&*[Binomial(n,Floor(n/j)): j in [1..n]]): n in [0..30]]; // _G. C. Greubel_, Feb 05 2024

%o (SageMath) [product(binomial(n,(n//j)) for j in range(1,n+1)) for n in range(31)] # _G. C. Greubel_, Feb 05 2024

%Y Cf. A010786, A034841, A051054, A092143, A272093.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Jun 20 2021