%I #21 Feb 08 2024 16:14:25

%S 0,0,2,18,204,2940,56670,1471806,52067512,2520298584,167850357210,

%T 15435027907530,1967345286257604,348527628228821652,

%U 86057693880611800438,29677160119074814383030,14321851348104417100842480

%N Number of cover relations summed over the rank-1 labeled posets on [n].

%C The rank of a poset is the number of cover relations in a maximal chain.

%C A cover relation in a poset is an ordered pair x <= y such that there is no z with x <= z <= y.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CoverRelation.html">Cover Relation</a>.

%F a(n) = Sum_{k=1..floor(n^2/4)} A052296(n,k)*k.

%t nn = 16; Table[Table[n!, {n, 0, nn}] CoefficientList[D[Series[Sum[Exp[y x]^Binomial[n, i]*Exp[ x]^(2^n - Binomial[n, i] - 1) x^n/n!, {n, 0, nn}], {x, 0, nn}], y] /. y -> 1, x]*i, {i, 1, nn - 1}] // Total

%Y Cf. A001831, A369919, A052296.

%K nonn

%O 0,3

%A _Geoffrey Critzer_, Feb 05 2024