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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.
6
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).
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
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Dec 22 2007
STATUS
approved