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A347843 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_5)^n.

%I A347843 #18 Feb 23 2025 17:27:11
%S A347843 1,1,7,249,44643,40065301,179833594207,4036127700341649,
%T A347843 452932494435315724443,254139954749268142006053901,
%U A347843 712988623255130761190069046824407,10001434425838325885839124865408303623049,701474672607858244757589244286886103482442884243
%N A347843 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_5)^n.
%H A347843 Alois P. Heinz, <a href="/A347843/b347843.txt">Table of n, a(n) for n = 0..54</a> (terms n = 1..40 from Álvar Ibeas)
%F A347843 a(n) = Sum_{L partition of n} A347488(n, L) * A036038(len(L), sig(L)), where sig(L) is the partition composed by the part multiplicities of L.
%F A347843 a(n) = Sum_{k=0..binomial(n,2)} 5^k * A381299(n,k). - _Alois P. Heinz_, Feb 21 2025
%e A347843 a(3) = 249 = 1 * 1 + 31 * 2 + 186 * 1, counting:
%e A347843 the unrefined chain 0 < (F_5)^3;
%e A347843 31 chains 0 < V < (F_5)^3, with dim(V) = 1; another
%e A347843 31 chains 0 < V < (F_5)^3, with dim(V) = 2; and
%e A347843 186 chains 0 < V_1 < V_2 < (F_5)^3.
%p A347843 b:= proc(o, u, t) option remember; `if`(u+o=0, 1, `if`(t=1,
%p A347843       b(u+o, 0$2), 0)+add(5^(u+j-1)*b(o-j, u+j-1, 1), j=1..o))
%p A347843     end:
%p A347843 a:= n-> b(n, 0$2):
%p A347843 seq(a(n), n=0..16);  # _Alois P. Heinz_, Feb 21 2025
%Y A347843 Cf. A289545, A347488, A036038, A381299.
%Y A347843 Column k=5 of A381426.
%K A347843 nonn
%O A347843 0,3
%A A347843 _Álvar Ibeas_, Sep 15 2021
%E A347843 a(0)=1 prepended by _Alois P. Heinz_, Feb 21 2025