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.

Sequence in context: A046595 A046587 A147563 this_sequence A088848 A088849 A138908

Adjacent sequences: A136210 A136211 A136212 this_sequence A136214 A136215 A136216

nonn,tabl

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