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

1, 1, 7, 249, 44643, 40065301, 179833594207, 4036127700341649, 452932494435315724443, 254139954749268142006053901, 712988623255130761190069046824407, 10001434425838325885839124865408303623049, 701474672607858244757589244286886103482442884243

Offset: 0

Author

Álvar Ibeas, Sep 15 2021

Keywords

• nonn

Examples

			a(3) = 249 = 1 * 1 + 31 * 2 + 186 * 1, counting:
the unrefined chain 0 < (F_5)^3;
31 chains 0 < V < (F_5)^3, with dim(V) = 1; another
31 chains 0 < V < (F_5)^3, with dim(V) = 2; and
186 chains 0 < V_1 < V_2 < (F_5)^3.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..54 (terms n = 1..40 from Álvar Ibeas)

Crossrefs

Cf. A289545, A347488, A036038, A381299.

Column k=5 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(5^(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..16);  # Alois P. Heinz, Feb 21 2025

Formula

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.

a(n) = Sum_{k=0..binomial(n,2)} 5^k * A381299(n,k). - Alois P. Heinz, Feb 21 2025

Extensions

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