OFFSET
0,8
COMMENTS
FORMULA
EXAMPLE
Triangle begins:
1;
1,1;
1,1,1;
1,3,1,1;
1,11,7,1,1;
1,59,63,15,1,1;
1,507,847,295,31,1,1;
1,7291,18319,8695,1271,63,1,1;
1,179835,664335,411447,78151,5271,127,1,1;
1,7735931,41472271,32564407,7702663,661479,21463,255,1,1;
1,587692667,4540159247,4429695671,1267673607,132944679,5440807,86615,511,1,1;
1,79670024827,884033807631,1056145981111,358144212999,44574850215,2207828071,44130215,347991,1023,1,1;
...
ILLUSTRATE RECURRENCE.
Set column 0 and main diagonal to all 1's; otherwise, for n>k>0:
T(n,k) = T(n-1,k-1) + Sum_{j=k..n-2} T(n-1,j)*2^j*T(j,k-1).
For example:
T(5,2) = T(4,1) + T(4,2)*2^2*T(2,1) + T(4,3)*2^3*T(3,1) = 63;
T(6,3) = T(5,2) + T(5,3)*2^3*T(3,2) + T(5,4)*2^4*T(4,2) = 295;
T(7,3) = T(6,2) + T(6,3)*2^3*T(3,2) + T(6,4)*2^4*T(4,2) + T(6,5)*2^5*T(5,2) = 8695.
...
ILLUSTRATE GENERATING METHOD using matrix powers of A152790.
Triangle A152790 begins:
1;
1,1;
-3,2,1;
84,-28,4,1;
-12520,3040,-240,8,1;
8233600,-1757824,103168,-1984,16,1;
-14411593728,4551192576,-235382784,3397632,-16128,32,1; ...
where g.f. of row n of A152790^(2^n) = (2^(2n-1) + y)*y^(n-1) for n>0.
For example, matrix power A152790^(2^7) begins:
1;
128,1;
15872,256,1;
2416640,61440,512,1;
444071936,16515072,229376,1024,1;
68048388096,3959422976,92274688,786432,2048,1;
137438953472,68719476736,8589934592,268435456,2097152,4096,1;
0,0,0,0,0,0,8192,1; <== g.f. of row 7 = (2^13 + y)*y^6
...
Now remove all factors of 2 from the first 6 rows of A152790^(2^7) to obtain:
1;
1, 1;
31, 1, 1;
295, 15, 1, 1;
847, 63, 7, 1, 1;
507, 59, 11, 3, 1, 1;
1, 1, 1, 1, 1, 1, 1.
Transposing this resultant triangle about the antidiagonal
yields the first 6 rows of this triangle A152795:
1;
1, 1;
1, 1, 1;
1, 3, 1, 1;
1, 11, 7, 1, 1;
1, 59, 63, 15, 1, 1;
1, 507, 847, 295, 31, 1, 1.
This process extracts the initial m-1 rows of this triangle from
the matrix power A152790^(2^m) for all m>1 -- a very pretty result!
PROG
(PARI) {T(n, k) = if(n==k || k==0, 1, T(n-1, k-1)+sum(j=k, n-2, T(n-1, j)*2^j*T(j, k-1)))}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(PARI) /* Define using matrix powers of A152790: */
{T(n, k) = my(P=Mat(1), W); if(n<k || k<0, 0, if(n==k || n==k+1 || k==0, 1, P=matrix(n, n, r, c, if(r>=c, T(r-1, c-1))); W=(matrix(n, n, r, c, P[n-c+1, n-r+1]*if(r==c, 1, 2^((r-1)*(r-2)-(c-1)^2+n))))^2; W[n-k+1, 1]/2^((n-k)*(n-k-1) + n+1)))}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Dec 19 2008
STATUS
approved