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.

1, 1, 5, 76, 785, 12181, 377708, 8009002, 171155505, 4073421919, 168532394115, 6213455777530, 198071252771780, 6383569557705276, 204582355050315856, 8766238064421938746, 446196770370016437201, 20584924968627941009331, 920598569147050035793061

Offset: 0

Author

Alois P. Heinz, Jul 18 2023

Keywords

• nonn

Examples

			a(2) = 5: 3abc1d, 3abd1c, 3acd1b, 3bcd1a, 22abcd.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450

Crossrefs

Cf. A364310.

Programs

Maple

b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*x^j*binomial(n, i*j), j=0..n/i))))
    end:
a:= n-> coeff(b(2*n$2), x, n):
seq(a(n), n=0..23);

Mathematica

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0,
   Sum[b[n - i*j, i - 1]*x^j*Binomial[n, i*j], {j, 0, n/i}]]]];
a[n_] := Coefficient[b[2n, 2n], x, n];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Nov 29 2023, from Maple code *)

Formula

a(n) = A364310(2n,n).