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A136328 a(n) = Wiener index of the odd graph O_n. 5
0, 3, 75, 1435, 25515, 436821, 7339332, 121782375, 2005392675, 32835436777, 535550923908, 8707954925033, 141270179732500, 2287544190032700, 36988236910737360, 597341791692978975, 9637351741503033075, 155353556752487795625, 2502545930175392062500 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The odd graph O_n (n>=2) is a graph whose vertices represent the (n-1)-subsets of {1,2,...,2n-1} and two vertices are connected if and only if they correspond to disjoint subsets. It is a distance regular graph. [Emeric Deutsch, Aug 20 2013]

REFERENCES

Kailasam Viswanathan Iyer, Some computational and graph theoretical aspects of Wiener index, Ph.D. Dissertation, Dept. of Comp. Sci. & Engg., National Institute of Technology, Trichy, India, 2007.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..200

R. Balakrishnan, N. Sridharan and K. Viswanathan Iyer, The Wiener index of odd graphs, J. Ind. Inst. Sci., vol. 86, no. 5, 2006.

R. Balakrishnan, N. Sridharan and K. Viswanathan Iyer,, The Wiener index of odd graphs, J. Ind. Inst. Sci., vol. 86, no. 5, 2006. [Cached copy]

Eric Weisstein's World of Mathematics, Odd Graph

Eric Weisstein's World of Mathematics, Wiener Index

FORMULA

a(n) = binomial(2*n - 1, n - 1)/2*(sum(((2*j + 1)*n)/(j + 1)*binomial(n - 1, j)^2, {j, 0, floor(n/2) - 1}) + sum((2*(n - 1 - j)*n)/(j + 1)*binomial(n - 1, j)^2, {j, floor(n/2), n - 2})). - Eric W. Weisstein, Sep 08 2017

A formula is "hidden" in the 2nd Maple program. B(n) and C(n) are the intersection arrays of O_n, H(n) is the Hosoya-Wiener polynomial of O_n, and Wi(n) is the Wiener index of O_n. - Emeric Deutsch, Aug 20 2013

a(n) = A301566(n)*binomial(2*n-1,n-1)/2. - Eric W. Weisstein, Mar 26 2018

EXAMPLE

a(2)=3 is the Wiener index of O_2 which is C_3.

a(3)=75 is the Wiener index of O_3 which is the Petersen graph.

MAPLE

A136328d := proc(k) add( (2*j+1)*binomial(k-1, j)^2/(1+j), j=0..(k/2-1) ); %+2*add( (k-1-j)*binomial(k-1, j)^2/(1+j), j=floor(k/2)..(k-2) ); k*% ; end proc:

A136328 := proc(n) binomial(2*n-1, n-1)*A136328d(n)/2 ; end proc: seq(A136328(n), n=1..20) ;

# R. J. Mathar, Sep 15 2010

B := proc (n) options operator, arrow: [seq(n-floor((1/2)*m), m = 1 .. n-1)] end proc: C := proc (n) options operator, arrow: [seq(ceil((1/2)*m), m = 1 .. n-1)] end proc: H := proc (n) options operator, arrow: (1/2)*binomial(2*n-1, n-1)*(sum((product(B(n)[r]/C(n)[r], r = 1 .. j))*t^j, j = 1 .. n-1)) end proc: Wi := proc (n) options operator, arrow: subs(t = 1, diff(H(n), t)) end proc: seq(Wi(n), n = 2 .. 20);

# Emeric Deutsch, Aug 20 2013

MATHEMATICA

Table[Binomial[2 n - 1, n - 1]/2 (Sum[((2 j + 1) n)/(j + 1) Binomial[n - 1, j]^2, {j, 0, Floor[n/2] - 1}] + Sum[(2 (n - 1 - j) n)/(j + 1) Binomial[n - 1, j]^2, {j, Floor[n/2], n - 2}]), {n, 20}] (* Eric W. Weisstein, Sep 08 2017 *)

Table[Sum[k Binomial[n, Ceiling[k/2]] Binomial[n - 1, Floor[k/2]] Binomial[2 n - 1, n - 1]/2, {k, n - 1}], {n, 20}] (* Eric W. Weisstein, Mar 27 2018 *)

PROG

(PARI) a(n) = sum(k=1, n-1, k*binomial(n, ceil(k/2))*binomial(n-1, k\2))*binomial(2*n-1, n-1)/2 \\ Andrew Howroyd, Mar 26 2018

CROSSREFS

Cf. A192835, A301566.

Sequence in context: A163131 A060869 A012491 * A003690 A230143 A228841

Adjacent sequences:  A136325 A136326 A136327 * A136329 A136330 A136331

KEYWORD

nonn

AUTHOR

K.V.Iyer, Mar 27 2008

EXTENSIONS

Extended by R. J. Mathar, Sep 15 2010

Terms a(18) and beyond from Andrew Howroyd, Mar 26 2018

STATUS

approved

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Last modified January 23 13:40 EST 2020. Contains 331171 sequences. (Running on oeis4.)