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, 9, 90, 836, 6856, 49787, 326618, 1977322, 11244976, 60908337, 317509874, 1605448440, 7920487752, 38297112551, 182108066522, 853884638758, 3956279351760, 18143381822941, 82466719670866, 371917534537524, 1665777832832136, 7415146800493139

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

Table of n, a(n) for n=0..21.

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

Adjacent sequences: A227710 A227711 A227712 * A227714 A227715 A227716

Keyword

nonn,easy

Author

Emeric Deutsch, Aug 07 2013

Status

approved