conjecture related to A027471

[Ross La Haye]

Let G be the Hasse diagram of a Boolean algebra of order n.  Define a matrix
M for G in the following manner --

M_ij = 0 if there is no (directed) walk from v_i to v_j of G.
M_ij = the length of the longest (directed) walk from v_i to v_j of G, if a
walk exists.

Summing all the entries in M by hand for n =0,1,2,3,4 I get 0,1,6,27,108
which seems to be Sum[Binomial(n,k)*k*2^(k-1),{k,0,n}] = n*3^(n-1) =
A027471(n+1).  I'm wondering if anyone can prove or disprove this...Thanks
in advance.

Ross


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