OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0.50 of the triangle, flattened
FORMULA
T(n, k) = A117418(n+k, 2*k). - G. C. Greubel, May 31 2021
EXAMPLE
Triangle begins:
1;
1, 1;
1, 3, 1;
1, 8, 5, 1;
1, 22, 20, 7, 1;
1, 65, 79, 37, 9, 1;
1, 208, 322, 180, 58, 11, 1;
1, 723, 1385, 871, 339, 83, 13, 1;
1, 2721, 6293, 4296, 1935, 550, 113, 15, 1;
1, 11053, 30152, 21821, 11092, 3465, 846, 148, 17, 1;
Column k of T equals column 2*k of A117418, which begins:
1;
1, 1;
1, 2, 1;
1, 4, 3, 1;
1, 9, 8, 4, 1;
1, 23, 22, 14, 5, 1;
1, 66, 65, 50, 20, 6, 1;
1, 209, 208, 191, 79, 28, 7, 1;
Let matrix R = SHIFT_RIGHT(A117418):
1;
0, 1;
0, 1, 1;
0, 1, 2, 1;
0, 1, 4, 3, 1;
0, 1, 9, 8, 4, 1;
0, 1, 23, 22, 14, 5, 1;
0, 1, 66, 65, 50, 20, 6, 1;
then the matrix product A117418*R yields this triangle.
MATHEMATICA
PROG
(PARI)
A117418(n, k) = if(n<k || k<0, 0, if(n==k || k==0, 1, if(n==k+1, n, sum(j=0, n-k, A117418(n-((k+1)\2), k\2+j)*A117418((k-1)\2+j, (k-1)\2)))));
for(n=0, 12, for(k=0, n, print1(A117425(n, k), ", "))) \\ modified by G. C. Greubel, May 31 2021
(Sage)
@CachedFunction
def A117418(n, k):
if (k<0 or k>n): return 0
elif (k==0 or k==n): return 1
elif (k==n-1): return n
flatten([[A117425(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 31 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Mar 14 2006
STATUS
approved