A136213 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

Offset

0,6

Comments

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

Links

Table of n, a(n) for n=0..69.

Formula

Column 0 forms the triple factorials A007559.

Example

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.

Prog

(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])}

Crossrefs

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

Sequence in context: A046595 A046587 A147563 * A088848 A088849 A251539

Adjacent sequences: A136210 A136211 A136212 * A136214 A136215 A136216

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 22 2007

Status

approved