A136225 Matrix square of triangle A136220, read by rows.

1, 2, 1, 8, 4, 1, 49, 26, 6, 1, 414, 232, 54, 8, 1, 4529, 2657, 629, 92, 10, 1, 61369, 37405, 9003, 1320, 140, 12, 1, 996815, 627435, 153276, 22606, 2385, 198, 14, 1, 18931547, 12248365, 3031553, 450066, 47500, 3904, 266, 16, 1, 412345688, 273211787

Offset: 0

Author

Paul D. Hanna, Jan 28 2008

Keywords

• nonn
• tabl

Comments

Column 0 of this triangle = column 1 of square array A136217.

Examples

			Let P = A136220, then this triangle is P^2 and begins:
1;
2, 1;
8, 4, 1;
49, 26, 6, 1;
414, 232, 54, 8, 1;
4529, 2657, 629, 92, 10, 1;
61369, 37405, 9003, 1320, 140, 12, 1;
996815, 627435, 153276, 22606, 2385, 198, 14, 1;
18931547, 12248365, 3031553, 450066, 47500, 3904, 266, 16, 1; ...
where column k of P^2 = column 0 of V^(k+1) and
triangle V = A136230 begins:
1;
2, 1;
8, 5, 1;
49, 35, 8, 1;
414, 325, 80, 11, 1;
4529, 3820, 988, 143, 14, 1;
61369, 54800, 14696, 2200, 224, 17, 1; ...
where column k of V = column 0 of P^(3k+2).
Triangle P = A136220 begins:
1;
1, 1;
3, 2, 1;
15, 10, 3, 1;
108, 75, 21, 4, 1;
1036, 753, 208, 36, 5, 1;
12569, 9534, 2637, 442, 55, 6, 1;
185704, 146353, 40731, 6742, 805, 78, 7, 1; ...
where column k of P = column 0 of U^(k+1) and U = A136228.
		

Crossrefs

Cf. columns: A136226, A136227; related tables: A136228 (U), A136230 (V), A136231 (W=P^3), A136217, A136218.

Programs

PARI

{T(n,k)=local(P=Mat(1),U,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]))));U=P*PShR^2; U=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,U[r,c], if(c==1,(P^3)[ #P,1],(P^(3*c-1))[r-c+1,1])))); P=matrix(#U, #U, r,c, if(r>=c, if(r<#R,P[r,c], (U^c)[r-c+1,1])))));(P^2)[n+1,k+1]}

Formula

Let P=A136220, V=A136230, then column k of P^2 (this triangle) = column 0 of V^(k+1) while column j of V = column 0 of P^(3j+2).