%I #16 Sep 08 2022 08:46:13

%S 0,1,0,36,15,6,2304,1280,704,384,250000,159375,101250,64125,40500,

%T 41990400,29113344,20155392,13934592,9621504,6635520,10169108964,

%U 7465417295,5476560950,4014772125,2941225000,2153396875,1575656250,3367254360064,2576980377600

%N Triangle read by rows: number of spanning trees obtained for an almost-complete bipartite graph by removing k disjoint edges from the complete bipartite graph K n,n with k<=n.

%H Fuji Zhang and Weigen Yan, <a href="http://dx.doi.org/10.1016/j.jcta.2008.10.004">Enumerating spanning trees of graphs with an involution</a>, Journal of Combinatorial Theory, Series A, Volume 116, Issue 3, April 2009, Pages 650-662 (see Theorem 4.1).

%F T(n, k) = ((n-2)*n+k)*(n-2)^(k-1)*n^(2*n-k-3).

%e Triangle begins:

%e 0;

%e 1, 0;

%e 36, 15, 6;

%e 2304, 1280, 704, 384;

%e 250000, 159375, 101250, 64125, 40500;

%e ...

%t Join[{0, 1, 0}, t[n_, k_]:=((n - 2) n + k) (n - 2)^(k - 1) n^(2 n - k - 3); Table[t[n, k], {n, 3, 10}, {k, n}]//Flatten] (* _Vincenzo Librandi_, Jul 24 2015 *)

%o (PARI) tabl(nn) = {for (n=1, nn, for (p=1, n, print1(((n-2)*n+p)*(n-2)^(p-1)*n^(2*n-p-3), ", ");); print(););}

%o (Magma) /* As triangle */ [[((n-2)*n+k)*(n-2)^(k-1)*n^(2*n-k-3): k in [1..n]]: n in [1.. 15]]; // _Vincenzo Librandi_, Jul 24 2015

%K nonn,tabl

%O 1,4

%A _Michel Marcus_, Jul 24 2015