A347841 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_3)^n.

1, 1, 5, 79, 3851, 567733, 251790113, 335313799327, 1340040415899803, 16067553466179577453, 577986341168068075687337, 62375143109859674070751394743, 20194282336027244435564571244298243, 19614041602745899032342581715038226919285

Offset: 0

Author

Álvar Ibeas, Sep 15 2021

Keywords

• nonn

Examples

			a(3) = 79 = 1 * 1 + 13 * 2 + 52 * 1, counting:
the unrefined chain 0 < (F_3)^3;
13 chains 0 < V < (F_3)^3, with dim(V) = 1; another
13 chains 0 < V < (F_3)^3, with dim(V) = 2; and
52 chains 0 < V_1 < V_2 < (F_3)^3.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..64 (terms n=1..40 from Álvar Ibeas)

Crossrefs

Cf. A289545, A347486, A036038, A381299.

Column k=3 of A381426.

Programs

Maple

b:= proc(o, u, t) option remember; `if`(u+o=0, 1, `if`(t=1,
      b(u+o, 0$2), 0)+add(3^(u+j-1)*b(o-j, u+j-1, 1), j=1..o))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..14);  # Alois P. Heinz, Feb 21 2025

Formula

a(n) = Sum_{L partition of n} A347486(n, L) * A036038(len(L), sig(L)), where sig(L) is the partition composed by the part multiplicities of L.

a(n) = Sum_{k=0..binomial(n,2)} 3^k * A381299(n,k). - Alois P. Heinz, Feb 21 2025

Extensions

a(0)=1 prepended by Alois P. Heinz, Feb 21 2025