_Paul D. Hanna (pauldhanna(AT)juno.com), _, Dec 22 2007

Triple factorial triangle, read by rows of 3n(n+1)/2+1 terms, where row n+1 is generated from row n by first inserting zeros in row n at positions {[m*(m+5)/6], m=0..3n-1} and then taking partial sums, starting with a '1' in row 0.

1, 1, 1, 1, 1, 4, 4, 4, 4, 3, 3, 2, 2, 1, 1, 28, 28, 28, 28, 24, 24, 20, 20, 16, 16, 12, 9, 9, 6, 4, 4, 2, 1, 1, 280, 280, 280, 280, 252, 252, 224, 224, 196, 196, 168, 144, 144, 120, 100, 100, 80, 64, 64, 48, 36, 27, 27, 18, 12, 8, 8, 4, 2, 1, 1, 3640, 3640, 3640, 3640, 3360

0,6

Square array A136212 is generated by a complementary process. This is the triple factorial variant of triangles A135877 (double factorials) and A127452 (factorials).

Column 0 forms the triple factorials A007559.

Triangle begins:

1;

1,1,1,1;

4,4,4,4,3,3,2,2,1,1;

28,28,28,28,24,24,20,20,16,16,12,9,9,6,4,4,2,1,1;

280,280,280,280,252,252,224,224,196,196,168,144,144,120,100,100,80,64,64,48,36,27,27,18,12,8,8,4,2,1,1;

3640,3640,3640,3640,3360,3360,3080,3080,2800,2800,2520,2268,2268,2016,1792,1792,1568,1372,1372,1176,1008,864,864,720,600,500,500,400,320,256,256,192,144,108,81,81,54,36,24,16,16,8,4,2,1,1;

...

To generate row 3, start with row 2:

[4,4,4,4,3,3,2,2,1,1];

insert zeros at positions [0,1,2,4,6,8,11,14,17] to get:

[0,0,0,4,0,4,0,4,0,4,3,0,3,2,0,2,1,0,1],

then take reverse partial sums (from right to left) to obtain row 3:

[28,28,28,28,24,24,20,20,16,16,12,9,9,6,4,4,2,1,1].

Continuing in this way will generate all the rows of this triangle.

(PARI) {T(n, k)=local(A=[1], B); if(n>0, for(i=1, n, m=1; B=[0, 0]; for(j=1, #A, if(j+m-1==(m*(m+7))\6, m+=1; B=concat(B, 0)); B=concat(B, A[j])); A=Vec(Polrev(Vec(Pol(B)/(1-x+O(x^#B))))))); if(k+1>#A, 0, A[k+1])}

Cf. A007559; related tables: A136212, A136218, A136214, A135877.

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 22 2007

approved