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A349703 Irregular triangle read by rows where T(n,k) is the number of free trees attaining the maximum terminal Wiener index (A349702) for a tree of n vertices among which k are leaves.

%I A349703 #12 Nov 28 2021 01:20:18
%S A349703 1,1,1,1,1,1,1,1,1,1,2,1,1,1,3,1,1,1,1,4,1,2,1,1,1,5,1,2,1,1,1,1,7,1,
%T A349703 3,1,2,1,1,1,8,1,3,1,2,1,1,1,1,10,1,4,1,3,1,2,1,1,1,12,1,4,1,3,1,2,1,
%U A349703 1,1,1,14,1,5,1,4,1,3,1,2,1,1,1,16,1,5,1,4,1,3,1,2,1,1,1
%N A349703 Irregular triangle read by rows where T(n,k) is the number of free trees attaining the maximum terminal Wiener index (A349702) for a tree of n vertices among which k are leaves.
%C A349703 Gutman, Furtula, and Petrović determine the maximum terminal Wiener index (A349702) possible in trees, and construct the trees which attain this maximum.
%C A349703 The triangle rows are all possible n,k combinations, which means k=n in rows n=0..2, and k=2..n-1 in rows n>=3.
%C A349703 For k even, a unique tree has the maximum index.
%C A349703 For k=3, all trees have the same index.
%H A349703 Kevin Ryde, <a href="/A349703/b349703.txt">Table of n, a(n) for n = 0..9593</a> (rows 0..140)
%H A349703 Ivan Gutman, Boris Furtula and Miroslav Petrović, <a href="https://doi.org/10.1007/s10910-008-9476-2">Terminal Wiener Index</a>, Journal of Mathematical Chemistry, volume 46, 2009, pages 522-531.
%F A349703 T(n,3) = A055290(n,3) = A001399(n-4) = round((n-1)^2 / 12).
%F A349703 T(n,k) = 1 for k even. [Gutman, Furtula, Petrović, theorem 4 (a)]
%F A349703 T(n,k) = ceiling((n-k)/2) for odd k >= 5. [Gutman, Furtula, Petrović, theorem 4 (b)]
%F A349703 G.f.: 1 + x*y + ( x^2*y^2 + ( x^4*y^3/(1-x^3) + x^5*y^4*(1+x*y-x^2)/(1-x^2*y^2) )/(1-x^2) )/(1-x).
%e A349703 Triangle begins
%e A349703       k=0  1  2  3  4  5  6  7  8
%e A349703   n=0;  1,
%e A349703   n=1;     1,
%e A349703   n=2;        1,
%e A349703   n=3;        1,
%e A349703   n=4;        1, 1,
%e A349703   n=5;        1, 1, 1,
%e A349703   n=6;        1, 2, 1, 1,
%e A349703   n=7;        1, 3, 1, 1, 1,
%e A349703   n=8;        1, 4, 1, 2, 1, 1,
%e A349703   n=9;        1, 5, 1, 2, 1, 1, 1,
%e A349703 For n=9,k=5, the T(9,5) = 2 trees are
%e A349703   *--*--*--*--*--*     *--*--*--*--*--*
%e A349703     /|         \         /   |      \
%e A349703    * *          *       *    *       *
%o A349703 (PARI) T(n,k) = if(n==1||k%2==0,1, k==3,(n-1)^2\/12, (n-k+1)>>1);
%Y A349703 Cf. A349702 (maximum index), A055290 (count all trees), A001399 (trees k=3 leaves).
%K A349703 easy,nonn,tabf
%O A349703 0,11
%A A349703 _Kevin Ryde_, Nov 26 2021