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A364323 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.

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