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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. 3
1, 2, 4, 3, 54, 27, 4, 408, 1152, 256, 5, 2500, 22500, 25000, 3125, 6, 13830, 315900, 988200, 583200, 46656, 7, 72030, 3709545, 25882780, 40588905, 14823774, 823543, 8, 360304, 39024384, 535754240, 1766195200, 1657012224, 411041792, 16777216 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Row sums give A014062, n >= 1.
Leading diagonal is A000312, n >= 1.
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
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..1830 (first 60 rows)
FORMULA
T(n,2) = binomial(n,2)*(binomial(2*n,n)-2). - Giovanni Resta, May 28 2015
EXAMPLE
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.
Triangle T(n,k) begins:
n\k 1 2 3 4 5 6 ...
1: 1
2: 2 4
3: 3 54 27
4: 4 408 1152 256
5: 5 2500 22500 25000 3125
6: 6 13830 315900 988200 583200 46656
...
n = 7: 7 72030 3709545 25882780 40588905 14823774 823543,
n = 8: 8 360304 39024384 535754240 1766195200 1657012224 411041792 16777216.
MATHEMATICA
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 *)
CROSSREFS
Sequence in context: A182103 A303053 A163089 * A111172 A173556 A247554
KEYWORD
nonn,tabl
AUTHOR
Adam J.T. Partridge, May 28 2015
STATUS
approved

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Last modified April 18 06:24 EDT 2024. Contains 371769 sequences. (Running on oeis4.)