A356292 Number of labeled trees on [n] that are centered.

1, 1, 0, 3, 4, 65, 726, 8617, 127688, 2374353, 50692330, 1198835561, 31297606572, 901114484569, 28449258421598, 976863784939785, 36199494609008656, 1438734246518372897, 61037354387458904274, 2753490065023053584713, 131645635680595606832180

Offset: 0

Author

Geoffrey Critzer, Aug 02 2022

Keywords

• nonn

Comments

This is the labeled version of A000676 which has the pertinent definitions.

Links

• J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
• Index entries for sequences related to trees

Crossrefs

Cf. A034854, A000676, A000272, A355671.

Programs

Mathematica

nn = 20; T = NestList[z Exp[#] &, z, nn]; G[k_, z_] := T[[k + 1]];H[k_, z_] := T[[k + 1]] - T[[k]];H[0, z_] := z; ReplacePart[ Sum[Range[0, nn]!CoefficientList[Series[G[m, z] (Exp[H[m, z]] - 1 - H[m, z]), {z, 0, nn}], z], {m, 0, nn/2 - 2}], {1 -> 1, 2 -> 1}]

Formula

a(n) = Sum_{d even} A034854(n,d).

a(n) = A000272(n) - A355671(n)