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

%I A347841 #20 Feb 23 2025 17:26:42
%S A347841 1,1,5,79,3851,567733,251790113,335313799327,1340040415899803,
%T A347841 16067553466179577453,577986341168068075687337,
%U A347841 62375143109859674070751394743,20194282336027244435564571244298243,19614041602745899032342581715038226919285
%N A347841 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_3)^n.
%H A347841 Alois P. Heinz, <a href="/A347841/b347841.txt">Table of n, a(n) for n = 0..64</a> (terms n=1..40 from Álvar Ibeas)
%F A347841 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.
%F A347841 a(n) = Sum_{k=0..binomial(n,2)} 3^k * A381299(n,k). - _Alois P. Heinz_, Feb 21 2025
%e A347841 a(3) = 79 = 1 * 1 + 13 * 2 + 52 * 1, counting:
%e A347841 the unrefined chain 0 < (F_3)^3;
%e A347841 13 chains 0 < V < (F_3)^3, with dim(V) = 1; another
%e A347841 13 chains 0 < V < (F_3)^3, with dim(V) = 2; and
%e A347841 52 chains 0 < V_1 < V_2 < (F_3)^3.
%p A347841 b:= proc(o, u, t) option remember; `if`(u+o=0, 1, `if`(t=1,
%p A347841       b(u+o, 0$2), 0)+add(3^(u+j-1)*b(o-j, u+j-1, 1), j=1..o))
%p A347841     end:
%p A347841 a:= n-> b(n, 0$2):
%p A347841 seq(a(n), n=0..14);  # _Alois P. Heinz_, Feb 21 2025
%Y A347841 Cf. A289545, A347486, A036038, A381299.
%Y A347841 Column k=3 of A381426.
%K A347841 nonn
%O A347841 0,3
%A A347841 _Álvar Ibeas_, Sep 15 2021
%E A347841 a(0)=1 prepended by _Alois P. Heinz_, Feb 21 2025