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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

Cf. A000312, A014062, A258152, A259051.

Sequence in context: A182103 A303053 A163089 * A111172 A173556 A247554

Adjacent sequences:  A258368 A258369 A258370 * A258372 A258373 A258374

KEYWORD

nonn,tabl

AUTHOR

Adam J.T. Partridge, May 28 2015

STATUS

approved

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Last modified June 21 06:55 EDT 2021. Contains 345358 sequences. (Running on oeis4.)