1,2

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 (mathar(AT)strw.leidenuniv.nl), Jul 26 2007

a(n,k)= [k*n-T(k-1)]!/[(k-1)*n-T(k-2)]! where T(n)=A000217(n). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 26 2007

In factorized notation the triangle starts

1;

1*2, 3;

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

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

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

which gives

1;

2, 3;

6, 20, 6;

24, 210, 72, 10;

120, 3024, 1320, 182, 15;

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

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 (mathar(AT)strw.leidenuniv.nl), Jul 26 2007

Cf. A093445, A093446, A093448.

Adjacent sequences: A093444 A093445 A093446 this_sequence A093448 A093449 A093450

Sequence in context: A121959 A075633 A124066 this_sequence A002078 A000372 A123930

nonn,tabl

Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 02 2004

More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 26 2007