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A093447 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).

%I A093447 #10 Aug 08 2015 19:59:44
%S A093447 1,2,3,6,20,6,24,210,72,10,120,3024,1320,182,15,720,55440,32760,4896,
%T A093447 380,21,5040,1235520,1028160,175560,13800,702,28,40320,32432400,
%U A093447 39070080,7893600,657720,32736,1190,36,362880,980179200,1744364160
%N A093447 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 A093447 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 A093447 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 A093447 In factorized notation the triangle starts
%e A093447 1;
%e A093447 1*2, 3;
%e A093447 1*2*3, 4*5, 6;
%e A093447 1*2*3*4, 5*6*7, 8*9, 10;
%e A093447 1*2*3*4*5, 6*7*8*9, 10*11*12, 13*14, 15;
%e A093447 which gives
%e A093447 1;
%e A093447 2, 3;
%e A093447 6, 20, 6;
%e A093447 24, 210, 72, 10;
%e A093447 120, 3024, 1320, 182, 15;
%e A093447 720,55440,32760, 4896, 380, 21;
%p A093447 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 A093447 Cf. A093445, A093446, A093448.
%K A093447 nonn,tabl
%O A093447 1,2
%A A093447 _Amarnath Murthy_, Apr 02 2004
%E A093447 More terms from _R. J. Mathar_, Jul 26 2007