OFFSET
0,5
COMMENTS
FORMULA
Column 0 equals the double factorials A001147(n) = (2n)!/(n!*2^n).
EXAMPLE
Triangle begins:
1;
1, 1, 1;
3, 3, 3, 2, 2, 1, 1;
15, 15, 15, 12, 12, 9, 9, 6, 4, 4, 2, 1, 1;
105, 105, 105, 90, 90, 75, 75, 60, 48, 48, 36, 27, 27, 18, 12, 8, 8, 4, 2, 1, 1;
945, 945, 945, 840, 840, 735, 735, 630, 540, 540, 450, 375, 375, 300, 240, 192, 192, 144, 108, 81, 81, 54, 36, 24, 16, 16, 8, 4, 2, 1, 1; ...
To generate the triangle, start with a single 1 in row 0,
and then obtain row n+1 from row n by inserting zeros
at positions [(m+2)^2/4 - 1] for m=1..2n+2 and then
taking reverse partial sums (i.e., summing from right to left).
Start with row 0, insert 2 zeros in front of the '1':
[0,0,1];
take reverse partial sums to get row 1:
[1,1,1];
insert zeros at positions [0,1,3,5]:
[0,0,1,0,1,0,1];
take reverse partial sums to get row 2:
[3,3,3,2,2,1,1];
insert zeros at positions [0,1,3,5,8,11]:
[0,0,3,0,3,0,3,2,0,2,1,0,1];
take reverse partial sums to get row 3:
[15,15,15,12,12,9,9,6,4,4,2,1,1];
insert zeros at positions [0,1,3,5,8,11,15,19]:
[0,0,15,0,15,0,15,12,0,12,9,0,9,6,4,0,4,2,1,0,1];
take reverse partial sums to get row 4:
[105,105,105,90,90,75,75,60,48,48,36,27,27,18,12,8,8,4,2,1,1].
PROG
(PARI) {T(n, k)=local(A=[1], B); if(n>0, for(i=1, n, m=1; B=[0]; for(j=1, #A, if(j+m-1==floor((m+2)^2/4)-1, 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])} /* for(n=0, 8, for(k=0, n*(n+1), print1(T(n, k), ", ")); print("")) */
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Dec 14 2007
STATUS
approved