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a(n) = lcm(1, 2, ..., n)^(n-1)/(n!*(n-1)!).
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%I #15 Jun 07 2023 09:11:36

%S 1,1,3,12,4500,9000,1512630000,1452124800000,111152892816000000,

%T 3112280998848000000,1849326140334157445511936000000,

%U 388358489470173063557506560000000,1607761625123067582500188167647056604083200000000

%N a(n) = lcm(1, 2, ..., n)^(n-1)/(n!*(n-1)!).

%C Consider a natural number n. Let t(k) denote the least common multiple (LCM) of {1, 2, ..., k} and Q(t(k)) denote the quotient of n when divided by t(k). Then the number M(n,k) of partitions of n with k parts can be expressed as a polynomial in Q(t(k)) with the leading coefficient (that is, the coefficient of Q(t(k))^(k-1)) c(k-1, k).

%H S. R. Park, J. Bae, H. G. Kang and I. Song, <a href="https://doi.org/10.1090/S0025-5718-07-02082-0">On the polynomial representation for the number of partitions with fixed length</a>, Mathematics of Computation, vol. 77, no. 262, pp. 1135-1151, 2008.

%t f[n_] := n (LCM @@ Range@n)^(n - 1)/n!^2; Array[f, 15] (* _Robert G. Wilson v_, May 30 2010 *)

%o (PARI) a(n) = lcm([1..n])^(n-1)/(n!*(n-1)!); \\ _Michel Marcus_, Jun 07 2023

%K nonn

%O 1,3

%A Iickho Song (i.song(AT)ieee.org), May 25 2010

%E a(9) onwards from _Robert G. Wilson v_, May 30 2010