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Triangle a(n,k) read by rows n which contain columns k=1,2,..,n, where each entry is the product of numbers (k-1)*n-T(k-2)+1 through k*n-T(k-1).
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%I #10 Aug 08 2015 19:59:44

%S 1,2,3,6,20,6,24,210,72,10,120,3024,1320,182,15,720,55440,32760,4896,

%T 380,21,5040,1235520,1028160,175560,13800,702,28,40320,32432400,

%U 39070080,7893600,657720,32736,1190,36,362880,980179200,1744364160

%N Triangle a(n,k) read by rows n which contain columns k=1,2,..,n, where each entry is the product of numbers (k-1)*n-T(k-2)+1 through k*n-T(k-1).

%C This is built by starting from the sequence 1,2,....,T(n) in row n, where T(n) is the triangular number A000217(n) and packaging its first n, the next n-1, the next n-2,... up to the last number in groups and writing down the product of each group in one cell of the triangle. The first column is A000142. The second column is essentially A006963. The 3rd column is essentially A001763. The diagonal is A000217. - _R. J. Mathar_, Jul 26 2007

%F a(n,k)= [k*n-T(k-1)]!/[(k-1)*n-T(k-2)]! where T(n)=A000217(n). - _R. J. Mathar_, Jul 26 2007

%e In factorized notation the triangle starts

%e 1;

%e 1*2, 3;

%e 1*2*3, 4*5, 6;

%e 1*2*3*4, 5*6*7, 8*9, 10;

%e 1*2*3*4*5, 6*7*8*9, 10*11*12, 13*14, 15;

%e which gives

%e 1;

%e 2, 3;

%e 6, 20, 6;

%e 24, 210, 72, 10;

%e 120, 3024, 1320, 182, 15;

%e 720,55440,32760, 4896, 380, 21;

%p A000217 := proc(n) n*(n+1)/2 ; end: A093447 := proc(n,k) factorial(k*n-A000217(k-1))/factorial((k-1)*n-A000217(k-2)) ; end: for n from 1 to 16 do for k from 1 to n do printf("%d, ",A093447(n,k)) ; od ; od: # _R. J. Mathar_, Jul 26 2007

%Y Cf. A093445, A093446, A093448.

%K nonn,tabl

%O 1,2

%A _Amarnath Murthy_, Apr 02 2004

%E More terms from _R. J. Mathar_, Jul 26 2007