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A224798
Triangle T, read by rows, where the matrix square T^2 results in shifting T right one column to drop the secondary diagonal (which consists of powers of 2).
4
1, 1, 1, 4, 2, 1, 24, 8, 4, 1, 224, 64, 16, 8, 1, 3200, 768, 192, 32, 16, 1, 70144, 15360, 2816, 640, 64, 32, 1, 2394112, 454656, 84992, 10752, 2304, 128, 64, 1, 127279104, 22528000, 3223552, 534528, 41984, 8704, 256, 128, 1, 10863804416, 1646198784, 247250944, 24158208, 3706880, 165888, 33792, 512, 256, 1
OFFSET
0,4
EXAMPLE
Triangle T begins:
1;
1, 1;
4, 2, 1;
24, 8, 4, 1;
224, 64, 16, 8, 1;
3200, 768, 192, 32, 16, 1;
70144, 15360, 2816, 640, 64, 32, 1;
2394112, 454656, 84992, 10752, 2304, 128, 64, 1;
127279104, 22528000, 3223552, 534528, 41984, 8704, 256, 128, 1;
10863804416, 1646198784, 247250944, 24158208, 3706880, 165888, 33792, 512, 256, 1; ...
Illustrate recurrence by products of row and column vectors:
T(4,1) = [16,8,1]*[1,4,16]~ = 16*1 + 8*4 + 1*16 = 64;
T(6,0) = [15360,2816,640,64,32,1]*[1,2,8,64,768,15360]~ = 70144;
T(7,1) = [84992,10752,2304,128,64,1]*[1,4,16,192,2816,84992]~ = 454656.
The matrix square T^2 begins:
1;
2, 1;
10, 4, 1;
72, 24, 8, 1;
768, 224, 64, 16, 1;
12288, 3200, 768, 192, 32, 1;
299008, 70144, 15360, 2816, 640, 64, 1;
11255808, 2394112, 454656, 84992, 10752, 2304, 128, 1;
664469504, 127279104, 22528000, 3223552, 534528, 41984, 8704, 256, 1;
62483333120, 10863804416, 1646198784, 247250944, 24158208, 3706880, 165888, 33792, 512, 1; ...
which equals T shifted right one column with the secondary diagonal dropped.
PROG
(PARI) T(n, k)=if(n<k || k<0, 0, if(n==k, 1, if(n==k+1, 2^(n-1), sum(j=k+1, n, T(n, j)*T(j, k+1) ))))
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
CROSSREFS
KEYWORD
tabl,nonn
AUTHOR
Paul D. Hanna, Apr 22 2013
STATUS
approved