A136230 Triangle V, read by rows, where column k of V^(j+1) = column j of P^(3k+2), for j>=0, k>=0 and where P=A136220.
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, 996815, 932761, 257264, 39468, 4123, 323, 20, 1, 18931547, 18426632, 5198680, 812801, 86506, 6919, 440, 23, 1, 412345688
Offset: 0
Examples
This triangle V 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; 996815, 932761, 257264, 39468, 4123, 323, 20, 1; 18931547, 18426632, 5198680, 812801, 86506, 6919, 440, 23, 1; ... where column k of V = column 0 of P^(3k+2) and 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; ... where column k of P^2 = column 0 of V^(k+1). Also, this triangle V equals the matrix product: V = P^2 * [P shift right one column] where P^2 = A136225 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; ... and P shift right one column begins: 1; 0, 1; 0, 1, 1; 0, 3, 2, 1; 0, 15, 10, 3, 1; 0, 108, 75, 21, 4, 1; 0, 1036, 753, 208, 36, 5, 1; ... Also, this triangle V equals the matrix product: V = U * [U shift down one row] where triangle U = A136228 begins: 1; 1, 1; 3, 4, 1; 15, 24, 7, 1; 108, 198, 63, 10, 1; 1036, 2116, 714, 120, 13, 1; ... and U shift down one row begins: 1; 1, 1; 1, 1, 1; 3, 4, 1, 1; 15, 24, 7, 1, 1; 108, 198, 63, 10, 1, 1; 1036, 2116, 714, 120, 13, 1, 1; ...
Crossrefs
Programs
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PARI
{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[n+1,k+1]}
Formula
Triangle W=P^3=A136231 transforms column k of V into column k+1 of V. This triangle equals the matrix products: V = P^2 * [P shift right one column] and V = U * [U shift down one row] (see examples).