A136227 -id:A136227 - cp's OEIS frontend

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

Paul D. Hanna, Jan 28 2008

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

			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.

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

{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]}

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).

A136234 Matrix square of triangle V = A136230, read by rows.

Original entry on oeis.org

1, 4, 1, 26, 10, 1, 232, 110, 16, 1, 2657, 1435, 248, 22, 1, 37405, 22135, 4240, 440, 28, 1, 627435, 397820, 81708, 9295, 686, 34, 1, 12248365, 8203057, 1773156, 214478, 17248, 986, 40, 1, 273211787, 191405232, 43039532, 5442349, 463267, 28747, 1340

Offset: 0

Paul D. Hanna, Feb 07 2008

			This triangle, V^2, begins:
1;
4, 1;
26, 10, 1;
232, 110, 16, 1;
2657, 1435, 248, 22, 1;
37405, 22135, 4240, 440, 28, 1;
627435, 397820, 81708, 9295, 686, 34, 1;
12248365, 8203057, 1773156, 214478, 17248, 986, 40, 1;
273211787, 191405232, 43039532, 5442349, 463267, 28747, 1340, 46, 1; ...
where column 0 of V^2 = column 1 of P^2 = triangle A136225.

Cf. A136227 (column 0); related tables: A136220 (P), A136228 (U), A136230 (V), A136231 (W=P^3), A136237 (V^3).

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

Column k of V^2 (this triangle) = column 1 of P^(3k+2), where P = triangle A136220.

cp's OEIS Frontend

A136225 Matrix square of triangle A136220, read by rows.

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