cp's OEIS Frontend

A136328 a(n) = Wiener index of the odd graph O_n.

%I A136328 #33 Feb 16 2025 08:33:07
%S A136328 0,3,75,1435,25515,436821,7339332,121782375,2005392675,32835436777,
%T A136328 535550923908,8707954925033,141270179732500,2287544190032700,
%U A136328 36988236910737360,597341791692978975,9637351741503033075,155353556752487795625,2502545930175392062500
%N A136328 a(n) = Wiener index of the odd graph O_n.
%C A136328 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]
%D A136328 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.
%H A136328 Andrew Howroyd, <a href="/A136328/b136328.txt">Table of n, a(n) for n = 1..200</a>
%H A136328 R. Balakrishnan, N. Sridharan and K. Viswanathan Iyer, <a href="http://journal.iisc.ernet.in/index.php/iisc/article/view/2323">The Wiener index of odd graphs</a>, J. Ind. Inst. Sci., vol. 86, no. 5, 2006.
%H A136328 R. Balakrishnan, N. Sridharan and K. Viswanathan Iyer,, <a href="/A136328/a136328.pdf">The Wiener index of odd graphs</a>, J. Ind. Inst. Sci., vol. 86, no. 5, 2006. [Cached copy]
%H A136328 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OddGraph.html">Odd Graph</a>
%H A136328 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>
%F A136328 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
%F A136328 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
%F A136328 a(n) = A301566(n)*binomial(2*n-1,n-1)/2. - _Eric W. Weisstein_, Mar 26 2018
%e A136328 a(2)=3 is the Wiener index of O_2 which is C_3.
%e A136328 a(3)=75 is the Wiener index of O_3 which is the Petersen graph.
%p A136328 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:
%p A136328 A136328 := proc(n) binomial(2*n-1,n-1)*A136328d(n)/2 ; end proc: seq(A136328(n),n=1..20) ;
%p A136328 # _R. J. Mathar_, Sep 15 2010
%p A136328 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);
%p A136328 # _Emeric Deutsch_, Aug 20 2013
%t A136328 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 *)
%t A136328 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 *)
%o A136328 (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
%Y A136328 Cf. A192835, A301566.
%K A136328 nonn
%O A136328 1,2
%A A136328 _K.V.Iyer_, Mar 27 2008
%E A136328 Extended by _R. J. Mathar_, Sep 15 2010
%E A136328 Terms a(18) and beyond from _Andrew Howroyd_, Mar 26 2018