%I #16 Nov 29 2023 05:59:52
%S 1,1,5,76,785,12181,377708,8009002,171155505,4073421919,168532394115,
%T 6213455777530,198071252771780,6383569557705276,204582355050315856,
%U 8766238064421938746,446196770370016437201,20584924968627941009331,920598569147050035793061
%N Number of partitions of 2n into n parts 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.
%H Alois P. Heinz, <a href="/A364323/b364323.txt">Table of n, a(n) for n = 0..450</a>
%F a(n) = A364310(2n,n).
%e a(2) = 5: 3abc1d, 3abd1c, 3acd1b, 3bcd1a, 22abcd.
%p b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
%p add(b(ni*j, i1)*x^j*binomial(n, i*j), j=0..n/i))))
%p end:
%p a:= n> coeff(b(2*n$2), x, n):
%p seq(a(n), n=0..23);
%t b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0,
%t Sum[b[n  i*j, i  1]*x^j*Binomial[n, i*j], {j, 0, n/i}]]]];
%t a[n_] := Coefficient[b[2n, 2n], x, n];
%t Table[a[n], {n, 0, 23}] (* _JeanFrançois Alcover_, Nov 29 2023, from Maple code *)
%Y Cf. A364310.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Jul 18 2023
