login
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.
3
1, 1, 4, 1, 18, 27, 1, 68, 288, 256, 1, 250, 2250, 5000, 3125, 1, 922, 15795, 65880, 97200, 46656, 1, 3430, 105987, 739508, 1932805, 2117682, 823543, 1, 12868, 696864, 7653632, 31539200, 59179008, 51380224, 16777216, 1, 48618, 4540968, 75687696, 461828790, 1320099444, 1919564892, 1377495072, 387420489
OFFSET
1,3
COMMENTS
This is the triangle A258371(n, m)/binomial(n, m).
For the corresponding partition array see A258152.
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..1830 (first 60 rows)
FORMULA
T(n, m) = sum over the A258152(n, k) entries corresponding to partitions of n with m parts; n >= 1, m = 1,2, ..., n.
T(n, m) = A258371(n, m)/binomial(n, m).
EXAMPLE
The triangle T(n, m) begins:
n\k 1 2 3 4 5 6 7
1: 1
2: 1 4
3: 1 18 27
4: 1 68 288 256
5: 1 250 2250 5000 3125
6: 1 922 15795 65880 97200 46656
7: 1 3430 105987 739508 1932805 2117682 823543
...
8: 1 12868 696864 7653632 31539200 59179008 51380224
16777216,
9: 1 48618 4540968 75687696 461828790 1320099444 1919564892 1377495072 387420489.
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).
MATHEMATICA
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 *)
CROSSREFS
Sequence in context: A201201 A077102 A258152 * A192722 A300141 A057968
KEYWORD
nonn,easy,tabl
AUTHOR
Wolfdieter Lang, Jun 18 2015
STATUS
approved