A152795 - OEIS

%I #6 Mar 29 2023 10:50:08

%S 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,

%T 7291,18319,8695,1271,63,1,1,1,179835,664335,411447,78151,5271,127,1,

%U 1,1,7735931,41472271,32564407,7702663,661479,21463,255,1,1,1,587692667

%N 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.

%C 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).

%F GENERATE FROM MATRIX POWERS OF A152790:

%F 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;

%F 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.

%e Triangle begins:

%e 1;

%e 1,1;

%e 1,1,1;

%e 1,3,1,1;

%e 1,11,7,1,1;

%e 1,59,63,15,1,1;

%e 1,507,847,295,31,1,1;

%e 1,7291,18319,8695,1271,63,1,1;

%e 1,179835,664335,411447,78151,5271,127,1,1;

%e 1,7735931,41472271,32564407,7702663,661479,21463,255,1,1;

%e 1,587692667,4540159247,4429695671,1267673607,132944679,5440807,86615,511,1,1;

%e 1,79670024827,884033807631,1056145981111,358144212999,44574850215,2207828071,44130215,347991,1023,1,1;

%e ...

%e ILLUSTRATE RECURRENCE.

%e Set column 0 and main diagonal to all 1's; otherwise, for n>k>0:

%e T(n,k) = T(n-1,k-1) + Sum_{j=k..n-2} T(n-1,j)*2^j*T(j,k-1).

%e For example:

%e T(5,2) = T(4,1) + T(4,2)*2^2*T(2,1) + T(4,3)*2^3*T(3,1) = 63;

%e T(6,3) = T(5,2) + T(5,3)*2^3*T(3,2) + T(5,4)*2^4*T(4,2) = 295;

%e 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.

%e ...

%e ILLUSTRATE GENERATING METHOD using matrix powers of A152790.

%e Triangle A152790 begins:

%e 1;

%e 1,1;

%e -3,2,1;

%e 84,-28,4,1;

%e -12520,3040,-240,8,1;

%e 8233600,-1757824,103168,-1984,16,1;

%e -14411593728,4551192576,-235382784,3397632,-16128,32,1; ...

%e where g.f. of row n of A152790^(2^n) = (2^(2n-1) + y)*y^(n-1) for n>0.

%e For example, matrix power A152790^(2^7) begins:

%e 1;

%e 128,1;

%e 15872,256,1;

%e 2416640,61440,512,1;

%e 444071936,16515072,229376,1024,1;

%e 68048388096,3959422976,92274688,786432,2048,1;

%e 137438953472,68719476736,8589934592,268435456,2097152,4096,1;

%e 0,0,0,0,0,0,8192,1; <== g.f. of row 7 = (2^13 + y)*y^6

%e ...

%e Now remove all factors of 2 from the first 6 rows of A152790^(2^7) to obtain:

%e 1;

%e 1, 1;

%e 31, 1, 1;

%e 295, 15, 1, 1;

%e 847, 63, 7, 1, 1;

%e 507, 59, 11, 3, 1, 1;

%e 1, 1, 1, 1, 1, 1, 1.

%e Transposing this resultant triangle about the antidiagonal

%e yields the first 6 rows of this triangle A152795:

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 1, 3, 1, 1;

%e 1, 11, 7, 1, 1;

%e 1, 59, 63, 15, 1, 1;

%e 1, 507, 847, 295, 31, 1, 1.

%e This process extracts the initial m-1 rows of this triangle from

%e the matrix power A152790^(2^m) for all m>1 -- a very pretty result!

%o (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)))}

%o for(n=0,10, for(k=0,n, print1(T(n,k),", "));print(""))

%o (PARI) /* Define using matrix powers of A152790: */

%o {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)))}

%o for(n=0,10, for(k=0,n, print1(T(n,k),", "));print(""))

%Y Cf. A152790; columns: A152796, A152797.

%K nonn,tabl

%O 0,8

%A _Paul D. Hanna_, Dec 19 2008