%I #11 Jul 20 2023 09:42:31
%S 1,1,7,81,841,10333,137677,1973401,29150551,484498301,8769443541,
%T 167200081777,3311785261513,66867027890601,1437872937193801,
%U 33031740883673521,796918495251727081,19807865344255857661,501642119664087055501,12828972405814319046601
%N Number of partitions of 2n with largest part n where each block of part i with multiplicity j is marked with a word of length i*j over a (2n)ary alphabet whose letters appear in alphabetical order and all 2n letters occur exactly once in the partition.
%C a(n) is also the number of endofunctions on [2n] such that n is the range maximum and the number of elements that are mapped to m is divisible by m. a(2) = 7: (2211), (2121), (2112), (1221), (1212), (1122), (2222).
%C All terms are odd.
%H Alois P. Heinz, <a href="/A364322/b364322.txt">Table of n, a(n) for n = 0..503</a>
%F a(n) = A364285(2n,n).
%e a(2) = 7: 2ab11cd, 2ac11bd, 2ad11bc, 2bc11ad, 2bd11ac, 2cd11ab, 22abcd.
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(b(ni*j, i1)*binomial(n, i*j), j=0..n/i)))
%p end:
%p a:= n> b(2*n, n)`if`(n=0, 0, b(2*n, n1)):
%p seq(a(n), n=0..23);
%Y Cf. A364285.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Jul 18 2023
