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Number of labeled trees on [n] that are bicentered.
2

%I #21 Aug 02 2022 23:51:25

%S 0,0,1,0,12,60,570,8190,134456,2408616,49307670,1159112130,

%T 30619757652,891045909468,28244653953698,969331283419590,

%U 35858099428919280,1423688804991442896,60402176709135347502,2726896792761748601226,130498364319404393167820

%N Number of labeled trees on [n] that are bicentered.

%C This is the labeled version of A000677 where pertinent definitions can be found.

%H J. Riordan, <a href="/A007401/a007401_8.pdf">The enumeration of trees by height and diameter</a>, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)

%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>

%F a(n) = Sum_{odd d} A034854(n,d).

%F a(n) = A000272(n) - A356292(n).

%t nn = 20; T = NestList[z Exp[#] &, z, nn]; G[k_, z_] := T[[k + 1]];

%t H[k_, z_] := T[[k + 1]] - T[[k]];H[0, z_] := z; ReplacePart[ Sum[Range[0, nn]! CoefficientList[Series[H[m, z]^2/2, {z, 0, nn}], z], {m, 1, nn/2 - 1}], 3 -> 1]

%Y Cf. A034854, A000677, A356292, A000272.

%K nonn

%O 0,5

%A _Geoffrey Critzer_, Aug 02 2022