1, 1, 1, 1, 1, 2, 4, 6, 6, 1, 3, 9, 24, 54, 90, 90, 1, 4, 16, 60, 204, 600, 1440, 2520, 2520, 1, 5, 25, 120, 540, 2220, 8100, 25200, 63000, 113400, 113400, 1, 6, 36, 210, 1170, 6120, 29520, 128520, 491400, 1587600, 4082400, 7484400, 7484400, 1, 7, 49, 336, 2226
Offset: 0
This triangle T begins:
1;
1, 1, 1;
1, 2, 4, 6, 6;
1, 3, 9, 24, 54, 90, 90;
1, 4, 16, 60, 204, 600, 1440, 2520, 2520;
1, 5, 25, 120, 540, 2220, 8100, 25200, 63000, 113400, 113400;
1, 6, 36, 210, 1170, 6120, 29520, 128520, 491400, 1587600, 4082400, 7484400, 7484400;
1, 7, 49, 336, 2226, 14070, 83790, 463680, 2346120, 10636920, 42071400, 139708800, 366735600, 681080400, 681080400,
1, 8, 64, 504, 3864, 28560, 201600, 1345680, 8401680, 48444480, 254016000, 1187524800, 4819953600, 16345929600, 43589145600, 81729648000, 81729648000,
1, 9, 81, 720, 6264, 52920, 430920, 3356640, 24811920, 172504080, 1116536400, 6646147200, 35835307200, 171632260800, 711047937600, 2451889440000, 6620101488000, 12504636144000, 12504636144000,
...
Rows 6 and 8 appear in Park (2015). - _N. J. A. Sloane_, Jan 31 2016
Let M be the triangular matrix that begins:
1;
1, 1;
1, 2, 1;
0, 3, 3, 1;
0, 0, 6, 4, 1;
0, 0, 0, 10, 5, 1; ...
where M(k+m,k) = C(k+m,k) for m=0,1,2 and zeros elsewhere.
Illustrate that row n of T = column 0 of M^n for n >= 0 as follows.
The matrix square M^2 begins:
1;
2, 1;
4, 4, 1;
6, 12, 6, 1;
6, 24, 24, 8, 1;
0, 30, 60, 40, 10, 1; ...
with column 0 of M^2 forming row 2 of T.
The matrix cube M^3 begins:
1;
3, 1;
9, 6, 1;
24, 27, 9, 1;
54, 96, 54, 12, 1;
90, 270, 240, 90, 15, 1;
90, 540, 810, 480, 135, 18, 1; ...
with column 0 of M^3 forming row 3 of T.
T(2,3)=6 because there are 6 ways to lodge 3 distinguishable balls, labeled by numbers 1,2 and 3, in 2 distinguishable boxes, each of which can hold at most 2 balls. - _N-E. Fahssi_, Apr 22 2009
T(5,8)=63000 because there are 63000 ways to assign 8 students to a dorm room when there are 5 different two-bed dorm rooms that are available. (See link for details of the count.) - _Dennis P. Walsh_, Feb 15 2011
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