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A227714 The hyper-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, 12, 178, 2688, 35995, 407992, 3952943, 33615105, 257526804, 1815863659, 11982128854, 74936243346, 448516091145, 2588488505682, 14488775962673, 79019272700951, 421449109356322, 2204733396548381, 11340131754176896, 57464577063754608 (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

Table of n, a(n) for n=0..19.

FORMULA

a(n) = -2026 -14671*n/12 -8077*n^2/24 -605*n^3/12 -83*n^4/24 +2^n*(2961/2 -9783*n/20 -1049*n^2/8 -147*n^3/8 +9*n^4/8 +21*n^5/40) +4^n*(1093/2-279*n/4 +51*n^2/2 -27*n^3/2 +9*n^4/4).

G.f.: (1 -25*x +364*x^2 -2888*x^3 +16604*x^4 -77320*x^5 +259299*x^6 -567034*x^7 +849760*x^8 -1145072*x^9 +1576816*x^10 -1535840*x^11 +730496*x^12 -160768*x^13 +83968*x^14 -8192*x^15) / ((1-x)^5*(1-2*x)^6*(1-4*x)^5).

MAPLE

a := proc (n) options operator, arrow: -2026-(14671/12)*n-(8077/24)*n^2-(605/12)*n^3-(83/24)*n^4+2^n*(2961/2-(9783/20)*n-(1049/8)*n^2-(147/8)*n^3+(9/8)*n^4+(21/40)*n^5)+4^n*(1093/2-(279/4)*n+(51/2)*n^2-(27/2)*n^3+(9/4)*n^4) end proc: seq(aa(n), n = 0 .. 25);

PROG

(MAGMA) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 -25*x +364*x^2 -2888*x^3 +16604*x^4 -77320*x^5 +259299*x^6 -567034*x^7 +849760*x^8 -1145072*x^9 +1576816*x^10 -1535840*x^11 +730496*x^12 -160768*x^13 +83968*x^14 -8192*x^15) / ((1-x)^5*(1-2*x)^6*(1-4*x)^5))); // Bruno Berselli, Aug 08 2013

CROSSREFS

Cf. A227713.

Sequence in context: A052208 A234531 A045952 * A130550 A073975 A069685

Adjacent sequences:  A227711 A227712 A227713 * A227715 A227716 A227717

KEYWORD

nonn,easy

AUTHOR

Emeric Deutsch, Aug 07 2013

STATUS

approved

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Last modified April 19 04:19 EDT 2019. Contains 322237 sequences. (Running on oeis4.)