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A347842 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_4)^n.

%I A347842 #20 Aug 07 2025 17:12:42
%S A347842 1,1,6,148,14808,5942608,9546508128,61355108818240,
%T A347842 1577381936031968640,162213856617581098030336,
%U A347842 66726795842176170072717129216,109792555585903911536355551233758208,722612693482570097701467493432061846673408,19023844570798442009810731239392846416136188284928
%N A347842 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_4)^n.
%H A347842 Alois P. Heinz, <a href="/A347842/b347842.txt">Table of n, a(n) for n = 0..58</a> (terms n = 1..40 from Álvar Ibeas)
%F A347842 a(n) = Sum_{L partition of n} A347487(n, L) * A036038(len(L), sig(L)), where sig(L) is the partition composed by the part multiplicities of L.
%F A347842 a(n) = Sum_{k=0..binomial(n,2)} 4^k * A381299(n,k). - _Alois P. Heinz_, Feb 21 2025
%e A347842 a(3) = 148 = 1 * 1 + 21 * 2 + 105 * 1, counting:
%e A347842 the unrefined chain 0 < (F_4)^3;
%e A347842 21 chains 0 < V < (F_4)^3, with dim(V) = 1; another
%e A347842 21 chains 0 < V < (F_4)^3, with dim(V) = 2; and
%e A347842 105 chains 0 < V_1 < V_2 < (F_4)^3.
%p A347842 b:= proc(o, u, t) option remember; `if`(u+o=0, 1, `if`(t=1,
%p A347842       b(u+o, 0$2), 0)+add(4^(u+j-1)*b(o-j, u+j-1, 1), j=1..o))
%p A347842     end:
%p A347842 a:= n-> b(n, 0$2):
%p A347842 seq(a(n), n=0..14);  # _Alois P. Heinz_, Feb 21 2025
%Y A347842 Cf. A289545, A347487, A036038, A381299.
%Y A347842 Column k=4 of A381426.
%K A347842 nonn
%O A347842 0,3
%A A347842 _Álvar Ibeas_, Sep 15 2021
%E A347842 a(0)=1 prepended by _Alois P. Heinz_, Feb 21 2025