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A258371 Triangle read by rows: T(n,k) is number of ways of arranging n indistinguishable points on an n X n square grid such that k rows contain at least one point.

%I A258371 #38 Nov 30 2016 22:04:36
%S A258371 1,2,4,3,54,27,4,408,1152,256,5,2500,22500,25000,3125,6,13830,315900,
%T A258371 988200,583200,46656,7,72030,3709545,25882780,40588905,14823774,
%U A258371 823543,8,360304,39024384,535754240,1766195200,1657012224,411041792,16777216
%N A258371 Triangle read by rows: T(n,k) is number of ways of arranging n indistinguishable points on an n X n square grid such that k rows contain at least one point.
%C A258371 Row sums give A014062, n >= 1.
%C A258371 Leading diagonal is A000312, n >= 1.
%C A258371 The triangle t(n,k) = T(n,k)/binomial(n,k) gives the number of ways to place n stones into the k X n grid of squares such that each of the k rows contains at least one stone. See A259051. One can use a partition array for this (and the T(n,k)) problem. See A258152. - _Wolfdieter Lang_, Jun 17 2015
%H A258371 Giovanni Resta, <a href="/A258371/b258371.txt">Table of n, a(n) for n = 1..1830</a> (first 60 rows)
%F A258371 T(n,2) = binomial(n,2)*(binomial(2*n,n)-2). - _Giovanni Resta_, May 28 2015
%e A258371 The number of ways of arranging eight pawns on a standard chessboard such that two rows contain at least one pawn is T(8,2)=360304.
%e A258371 Triangle T(n,k) begins:
%e A258371 n\k 1      2        3       4        5       6 ...
%e A258371 1:  1
%e A258371 2:  2      4
%e A258371 3:  3     54      27
%e A258371 4:  4    408    1152      256
%e A258371 5:  5   2500   22500    25000     3125
%e A258371 6:  6  13830  315900   988200   583200   46656
%e A258371 ...
%e A258371 n = 7:  7  72030 3709545 25882780  40588905 14823774 823543,
%e A258371 n = 8:  8 360304 39024384 535754240 1766195200 1657012224 411041792 16777216.
%t A258371 T[n_,k_]:= Binomial[n,k] * Sum[Multinomial@@ (Last/@ Tally[e]) * Times@@ Binomial[n,e], {e, IntegerPartitions[n, {k}]}]; Flatten@ Table[ T[n,k],{n,9}, {k,n}] (* _Giovanni Resta_, May 28 2015 *)
%Y A258371 Cf. A000312, A014062, A258152, A259051.
%K A258371 nonn,tabl
%O A258371 1,2
%A A258371 _Adam J.T. Partridge_, May 28 2015