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Triangle T(n,m) for the number of ways to put n stones into an m X n square grid such that each of the m rows contains at least one stone.
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%I #19 Jun 29 2015 12:20:19

%S 1,1,4,1,18,27,1,68,288,256,1,250,2250,5000,3125,1,922,15795,65880,

%T 97200,46656,1,3430,105987,739508,1932805,2117682,823543,1,12868,

%U 696864,7653632,31539200,59179008,51380224,16777216,1,48618,4540968,75687696,461828790,1320099444,1919564892,1377495072,387420489

%N Triangle T(n,m) for the number of ways to put n stones into an m X n square grid such that each of the m rows contains at least one stone.

%C This is the triangle A258371(n, m)/binomial(n, m).

%C For the corresponding partition array see A258152.

%H Giovanni Resta, <a href="/A259051/b259051.txt">Table of n, a(n) for n = 1..1830</a> (first 60 rows)

%F T(n, m) = sum over the A258152(n, k) entries corresponding to partitions of n with m parts; n >= 1, m = 1,2, ..., n.

%F T(n, m) = A258371(n, m)/binomial(n, m).

%e The triangle T(n, m) begins:

%e n\k 1 2 3 4 5 6 7

%e 1: 1

%e 2: 1 4

%e 3: 1 18 27

%e 4: 1 68 288 256

%e 5: 1 250 2250 5000 3125

%e 6: 1 922 15795 65880 97200 46656

%e 7: 1 3430 105987 739508 1932805 2117682 823543

%e ...

%e 8: 1 12868 696864 7653632 31539200 59179008 51380224

%e 16777216,

%e 9: 1 48618 4540968 75687696 461828790 1320099444 1919564892 1377495072 387420489.

%e a(4, 2) = 68 from the sum 32 + 36 of the n=4 row of A258152 which belong to the partitions of 4 with m=2 parts, namely (1, 3) and (2, 2).

%t T[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_, Jun 18 2015 *)

%Y Cf. A258152, A258371.

%K nonn,easy,tabl

%O 1,3

%A _Wolfdieter Lang_, Jun 18 2015