A135897 Triangle, read by rows, equal to R^4, the matrix 4th power of R = A135894.

1, 4, 1, 26, 12, 1, 216, 138, 20, 1, 2171, 1716, 330, 28, 1, 25628, 23647, 5440, 602, 36, 1, 348050, 362116, 94515, 12348, 954, 44, 1, 5352788, 6138746, 1761940, 258391, 23400, 1386, 52, 1, 92056223, 114543428, 35429974, 5662412, 572331, 39556, 1898

Offset: 0

Author

Paul D. Hanna, Dec 15 2007

Keywords

• nonn
• tabl

Comments

Triangle P = A135880 is defined by: column k of P^2 equals column 0 of P^(2k+2) such that column 0 of P^2 equals column 0 of P shift left.

Examples

			Triangle R^4 begins:
1;
4, 1;
26, 12, 1;
216, 138, 20, 1;
2171, 1716, 330, 28, 1;
25628, 23647, 5440, 602, 36, 1;
348050, 362116, 94515, 12348, 954, 44, 1;
5352788, 6138746, 1761940, 258391, 23400, 1386, 52, 1; ...
where R = A135894 begins:
1;
1, 1;
2, 3, 1;
6, 12, 5, 1;
25, 63, 30, 7, 1;
138, 421, 220, 56, 9, 1;
970, 3472, 1945, 525, 90, 11, 1; ...
where column k of R = column 0 of P^(2k+1)
and P = A135880 begins:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1; ...
where column k of P equals column 0 of R^(k+1).
		

Crossrefs

Cf. A135884 (column 0); A135894 (R), A135880 (P), A135888 (P^3), A135892 (P^5).

Programs

PARI

{T(n,k)=local(P=Mat(1),R=Mat(1),PShR);if(n>0,for(i=0,n, PShR=matrix(#P,#P, r,c, if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1]))));R=P*PShR; R=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,R[r,c], if(c==1,(P^2)[ #P,1],(P^(2*c-1))[r-c+1,1])))); P=matrix(#R, #R, r,c, if(r>=c, if(r<#R,P[r,c], (R^c)[r-c+1,1])))));(R^4)[n+1,k+1]}

Formula

Column k of R^4 = column 3 of P^(2k+1) for k>=0 where triangle P = A135880; column 0 of R^4 = column 3 of P; column 1 of R^4 = column 3 of P^3; column 2 of R^4 = column 3 of P^5.