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A227713 The Wiener index of the tree g[n] (n>=0) defined recursively in the following manner: denoting by P[n] the path on n vertices, we define g[0] =P[2] while g[n] (n>=1) is the tree obtained by identifying the roots of 2 copies of g[n-1] and one of the end-vertices of P[n+1]; the root of g[n] is defined to be the other end-vertex of P[n+1]. 1
1, 9, 90, 836, 6856, 49787, 326618, 1977322, 11244976, 60908337, 317509874, 1605448440, 7920487752, 38297112551, 182108066522, 853884638758, 3956279351760, 18143381822941, 82466719670866, 371917534537524, 1665777832832136, 7415146800493139 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Roughly speaking, g[4], for example, is obtained from the planted full binary tree of height 5 by replacing the edges at the levels 1,2,3,4 with paths of lengths 4, 3, 2, and 1, respectively.
The value of a(4) has been checked by the direct evaluation of the Wiener index (using Maple).
LINKS
Index entries for linear recurrences with constant coefficients, signature (24,-254,1564,-6225,16820,-31496,40896,-36112,20672,-6912,1024).
FORMULA
a(n) = 80 +128*n/3 +19*n^2/2 +5*n^3/6 +2^n*(-64 +38*n +21*n^2/2 +3*n^3/2) +4^n*(-15 -27*n/2 +9*n^2/2).
G.f.: (1 -15*x +128*x^2 -602*x^3 +1801*x^4 -3968*x^5 +6016*x^6 -5528*x^7 +3120*x^8 -1344*x^9 +256*x^10) / ((1-x)^4*(1-2*x)^4*(1-4*x)^3).
EXAMPLE
a(1)=9 because g[1] is the tree in the shape of Y and 3*1 + 3*2 = 9.
MAPLE
a := proc (n) options operator, arrow: 80+(128/3)*n+(19/2)*n^2+(5/6)*n^3 +2^n*(-64+38*n +(21/2)*n^2+(3/2)*n^3)+4^n*(-15-(27/2)*n+(9/2)*n^2) end proc: seq(a(n), n = 0 .. 25);
PROG
(Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-15*x +128*x^2-602*x^3 +1801*x^4-3968*x^5+6016*x^6 -5528*x^7+3120*x^8 -1344*x^9 +256*x^10)/((1-x)^4*(1-2*x)^4*(1-4*x)^3))); // Bruno Berselli, Aug 08 2013
CROSSREFS
Cf. A227714.
Sequence in context: A158609 A229250 A242161 * A343366 A057092 A156577
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Aug 07 2013
STATUS
approved

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Last modified March 28 11:59 EDT 2024. Contains 371254 sequences. (Running on oeis4.)