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A356375 Number of unlabeled centered trees with n nodes that have exactly one diametral path (up to direction of traversal). 0
0, 1, 0, 1, 0, 1, 2, 5, 9, 21, 44, 107, 247, 607, 1465, 3649, 9087, 23059, 58831, 151832, 394074, 1030492, 2708343, 7157735, 19002282, 50676945, 135691504, 364725995, 983775878, 2662271414, 7226368722, 19670528467, 53685042694, 146879757368, 402786655780, 1106968400532 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
COMMENTS
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.
LINKS
J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
MATHEMATICA
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[
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]
CROSSREFS
Cf. A000676.
Sequence in context: A117186 A155042 A001851 * A303963 A304349 A024822
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Aug 04 2022
STATUS
approved

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Last modified May 26 13:25 EDT 2024. Contains 372826 sequences. (Running on oeis4.)