A355671 Number of labeled trees on [n] that are bicentered.

0, 0, 1, 0, 12, 60, 570, 8190, 134456, 2408616, 49307670, 1159112130, 30619757652, 891045909468, 28244653953698, 969331283419590, 35858099428919280, 1423688804991442896, 60402176709135347502, 2726896792761748601226, 130498364319404393167820

Offset: 0

Author

Geoffrey Critzer, Aug 02 2022

Keywords

• nonn

Comments

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

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, A000677, A356292, A000272.

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[H[m, z]^2/2, {z, 0, nn}], z], {m, 1, nn/2 - 1}], 3 -> 1]

Formula

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

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