A267849 - OEIS

%I #27 May 04 2017 06:03:16

%S 1,1,3,1,6,12,1,9,36,60,1,12,72,240,360,1,15,120,600,1800,2520,1,18,

%T 180,1200,5400,15120,20160,1,21,252,2100,12600,52920,141120,181440,1,

%U 24,336,3360,25200,141120,564480,1451520,1814400,1,27,432,5040,45360,317520,1693440,6531840,16329600,19958400

%N Triangular array: T(n,k) is the 2-row creation rook number to place k rooks on a 3 x n board.

%C T(n,k) is the number of ways to place k rooks in a 3 x n Ferrers board (or diagram) under the Goldman-Haglund i-row creation rook mode for i=2. All row heights are 3.

%H Jay Goldman and James Haglund, <a href="http://dx.doi.org/10.1006/jcta.2000.3113">Generalized rook polynomials</a>, J. Combin. Theory A 91 (2000), 509-530.

%F T(n,k) = T(n-1,k) + (k+2) T(n-1,k-1) subject to T(0,0)=1, T(n,k)=0 for n<k and for k<0.

%e The triangle T(n,k) begins in row n=0 with columns 0<=k<=n:

%e      1

%e      1      3

%e      1      6     12

%e      1      9     36     60

%e      1     12     72    240    360

%e      1     15    120    600   1800   2520

%e      1     18    180   1200   5400  15120  20160

%e      1     21    252   2100  12600  52920 141120 181440

%e      1     24    336   3360  25200 141120 564480 1451520 1814400

%e      1     27    432   5040  45360 317520 1693440 6531840 16329600 19958400

%Y Cf. A013610 (1-rook coefficients on the 3xn board), A121757 (2-rook coeffs. on the 2xn board), A013609 (1-rook coeffs. on the 2xn board), A013611 (1-rook coeffs. on the 4xn board), A008279 (2-rook coeffs. on the 1xn board), A082030 (row sums?), A049598 (column k=2), A007531 (column k=3 w/o factor 10), A001710 (diagonal?).

%K nonn,easy,tabl

%O 0,3

%A _Ken Joffaniel M. Gonzales_, Jan 21 2016

%E Triangle simplified (reversing rows, offset 0). - _R. J. Mathar_, May 03 2017