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%I #3 Mar 30 2012 18:37:08
%S 1,3,1,15,6,1,108,48,9,1,1036,495,99,12,1,12569,6338,1323,168,15,1,
%T 185704,97681,21036,2754,255,18,1,3247546,1767845,390012,52204,4950,
%U 360,21,1,65762269,36839663,8287041,1128404,108860,8073,483,24,1,1515642725
%N Triangle W, read by rows, where column k of W = column 0 of W^(k+1) for k>=0 such that W equals the matrix cube of P = A136220 with column 0 of W = column 0 of P shift up one row.
%C This triangle W is the column transform for triangles U=A136228 and V=A136230: W * [column k of U] = column k+1 of U and W * [column k of V] = column k+1 of V, for k>=0.
%e Triangle W begins:
%e 1;
%e 3, 1;
%e 15, 6, 1;
%e 108, 48, 9, 1;
%e 1036, 495, 99, 12, 1;
%e 12569, 6338, 1323, 168, 15, 1;
%e 185704, 97681, 21036, 2754, 255, 18, 1;
%e 3247546, 1767845, 390012, 52204, 4950, 360, 21, 1;
%e 65762269, 36839663, 8287041, 1128404, 108860, 8073, 483, 24, 1; ...
%e where column k of W = column 0 of W^(k+1) such that W = P^3
%e and 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^3 = column 0 of P^(3k+3) such that
%e column 0 of P^3 = column 0 of P shift up one row.
%e Also, this triangle W equals the matrix product:
%e W = V * [V shift down one row]
%e where triangle V = A136230 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 and V shift down one row begins:
%e 1;
%e 1, 1;
%e 2, 1, 1;
%e 8, 5, 1, 1;
%e 49, 35, 8, 1, 1;
%e 414, 325, 80, 11, 1, 1;
%e 4529, 3820, 988, 143, 14, 1, 1; ...
%o (PARI) {T(n,k)=local(P=Mat(1),U=Mat(1),W=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; 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]))); W=P^3;));W[n+1,k+1]}
%Y Cf. A136221 (column 0); related tables: A136220 (P), A136225 (P^2), A136228 (U), A136230 (V), A136235 (W^2), A136238 (W^3); A136217, A136218.
%K nonn,tabl
%O 0,2
%A _Paul D. Hanna_, Jan 28 2008
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