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A356375 Number of unlabeled centered trees with n nodes that have exactly one diametral path (up to direction of traversal).

%I A356375 #8 Aug 04 2022 15:55:22
%S A356375 0,1,0,1,0,1,2,5,9,21,44,107,247,607,1465,3649,9087,23059,58831,
%T A356375 151832,394074,1030492,2708343,7157735,19002282,50676945,135691504,
%U A356375 364725995,983775878,2662271414,7226368722,19670528467,53685042694,146879757368,402786655780,1106968400532
%N A356375 Number of unlabeled centered trees with n nodes that have exactly one diametral path (up to direction of traversal).
%C A356375 A diametral path in a tree is a path of maximum length. A diametral path in a centered tree is necessarily of even length.  Its endpoints are leaves and its middle point is the center of the tree.  A centered tree with exactly one diametral path of length 2m can be decomposed into a rooted tree of height at most m-1 along with exactly 2 rooted trees of height exactly m-1.  It appears that almost all centered trees (A000676) have exactly one diametral path.
%H A356375 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 A356375 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%t A356375 nn = 35; S[0, x_] := x; S[k_, x_] := Total[Nest[CoefficientList[Series[Product[1/(1 - x^i)^#[[i]], {i, 1, Length[#]}], {x, 0, nn}], x] &, {1}, k] Table[x^i, {i, 1, nn + 1}]]; R[0, x] := x;R[k_, x_] := S[k, x] - S[k - 1, x]; ReplacePart[ Sum[PadRight[
%t A356375    CoefficientList[Series[S[m, x] (R[m, x]^2 + (R[m, x] /. x -> x^2))/2, {x, 0, nn}],x], nn + 1], {m, 0, nn/2}], 2 -> 1]
%Y A356375 Cf. A000676.
%K A356375 nonn
%O A356375 0,7
%A A356375 _Geoffrey Critzer_, Aug 04 2022