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A259051 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.

%I A259051 #19 Jun 29 2015 12:20:19
%S A259051 1,1,4,1,18,27,1,68,288,256,1,250,2250,5000,3125,1,922,15795,65880,
%T A259051 97200,46656,1,3430,105987,739508,1932805,2117682,823543,1,12868,
%U A259051 696864,7653632,31539200,59179008,51380224,16777216,1,48618,4540968,75687696,461828790,1320099444,1919564892,1377495072,387420489
%N A259051 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 A259051 This is the triangle A258371(n, m)/binomial(n, m).
%C A259051 For the corresponding partition array see A258152.
%H A259051 Giovanni Resta, <a href="/A259051/b259051.txt">Table of n, a(n) for n = 1..1830</a> (first 60 rows)
%F A259051 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 A259051 T(n, m) = A258371(n, m)/binomial(n, m).
%e A259051 The triangle T(n, m) begins:
%e A259051 n\k 1     2       3        4         5         6           7
%e A259051 1:  1
%e A259051 2:  1     4
%e A259051 3:  1    18      27
%e A259051 4:  1    68     288      256
%e A259051 5:  1   250    2250     5000      3125
%e A259051 6:  1   922   15795    65880     97200      46656
%e A259051 7:  1  3430  105987   739508   1932805    2117682     823543
%e A259051 ...
%e A259051 8:  1 12868  696864  7653632  31539200 59179008 51380224
%e A259051 16777216,
%e A259051 9:  1 48618 4540968 75687696 461828790 1320099444 1919564892 1377495072 387420489.
%e A259051 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 A259051 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 A259051 Cf. A258152, A258371.
%K A259051 nonn,easy,tabl
%O A259051 1,3
%A A259051 _Wolfdieter Lang_, Jun 18 2015