A152795 Triangle defined by T(n,k) = T(n-1,k-1) + Sum_{j=k..n-2} T(n-1,j)*2^j*T(j,k-1) for n>k>0 with T(n,n)=T(n,0)=1, read by rows.

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

Offset

0,8

Comments

This triangle may also be generated by matrix powers of triangle A152790 which satisfies: g.f. of row n of A152790^(2^n) = (2^(2n-1) + y)*y^(n-1) for n>0 (see example).

Links

Table of n, a(n) for n=0..56.

Formula

GENERATE FROM MATRIX POWERS OF A152790:

T(n,k) = [A152790^(2^(n+1))](n-k,0) / 2^((n-k)*(n-k-1)+n+1) for n>k, with T(n,n)=1;

T(n,k) = [A152790^(2^(n+m))](n-k+m-1,m-1) / 2^((n-k+m-1)*(n-k+m-2) - (m-1)^2 + n+m) for m>0, n>k, with T(n,n)=1.

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

Cf. A152790; columns: A152796, A152797.

Sequence in context: A228637 A352431 A362783 * A338817 A121585 A261959

Adjacent sequences: A152792 A152793 A152794 * A152796 A152797 A152798

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 19 2008

Status

approved