%I #3 Mar 30 2012 18:37:08
%S 1,2,1,8,5,1,49,35,8,1,414,325,80,11,1,4529,3820,988,143,14,1,61369,
%T 54800,14696,2200,224,17,1,996815,932761,257264,39468,4123,323,20,1,
%U 18931547,18426632,5198680,812801,86506,6919,440,23,1,412345688
%N 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.
%F 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).
%e This triangle V begins:
%e 1;
%e 2, 1;
%e 8, 5, 1;
%e 49, 35, 8, 1;
%e 414, 325, 80, 11, 1;
%e 4529, 3820, 988, 143, 14, 1;
%e 61369, 54800, 14696, 2200, 224, 17, 1;
%e 996815, 932761, 257264, 39468, 4123, 323, 20, 1;
%e 18931547, 18426632, 5198680, 812801, 86506, 6919, 440, 23, 1; ...
%e where column k of V = column 0 of P^(3k+2) and
%e triangle P = A136220 begins:
%e 1;
%e 1, 1;
%e 3, 2, 1;
%e 15, 10, 3, 1;
%e 108, 75, 21, 4, 1;
%e 1036, 753, 208, 36, 5, 1;
%e 12569, 9534, 2637, 442, 55, 6, 1; ...
%e where column k of P^2 = column 0 of V^(k+1).
%e Also, this triangle V equals the matrix product:
%e V = P^2 * [P shift right one column]
%e where P^2 = A136225 begins:
%e 1;
%e 2, 1;
%e 8, 4, 1;
%e 49, 26, 6, 1;
%e 414, 232, 54, 8, 1;
%e 4529, 2657, 629, 92, 10, 1;
%e 61369, 37405, 9003, 1320, 140, 12, 1; ...
%e and P shift right one column begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 3, 2, 1;
%e 0, 15, 10, 3, 1;
%e 0, 108, 75, 21, 4, 1;
%e 0, 1036, 753, 208, 36, 5, 1; ...
%e Also, this triangle V equals the matrix product:
%e V = U * [U shift down one row]
%e where triangle U = A136228 begins:
%e 1;
%e 1, 1;
%e 3, 4, 1;
%e 15, 24, 7, 1;
%e 108, 198, 63, 10, 1;
%e 1036, 2116, 714, 120, 13, 1; ...
%e and U shift down one row begins:
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 3, 4, 1, 1;
%e 15, 24, 7, 1, 1;
%e 108, 198, 63, 10, 1, 1;
%e 1036, 2116, 714, 120, 13, 1, 1; ...
%o (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]}
%Y Cf. A136226 (column 0), A136229 (column 1); related tables: A136220 (P), A136225 (P^2), A136230 (V), A136231 (W=P^3), A136234 (V^2), A136237 (V^3); A136217, A136218.
%K nonn,tabl
%O 0,2
%A _Paul D. Hanna_, Jan 28 2008