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

Author

Paul D. Hanna, Dec 22 2007

Keywords

• nonn
• tabl

Comments

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

Examples

			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.
		

Crossrefs

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

Programs

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])}

Formula

Column 0 forms the triple factorials A007559.