The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

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