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%I #2 Mar 30 2012 18:37:08
%S 1,0,1,0,1,1,0,2,2,1,0,6,7,3,1,0,25,34,15,4,1,0,138,215,99,26,5,1,0,
%T 970,1698,814,216,40,6,1,0,8390,16220,8057,2171,400,57,7,1,0,86796,
%U 182714,93627,25628,4740,666,77,8,1,0,1049546,2378780,1252752,348050,64805
%N Triangle, read by rows equal to the matrix product P^-1*R, where P = A135880 and R = A135894; P^-1*R equals triangle P shifted right one column.
%e Triangle begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 2, 2, 1;
%e 0, 6, 7, 3, 1;
%e 0, 25, 34, 15, 4, 1;
%e 0, 138, 215, 99, 26, 5, 1;
%e 0, 970, 1698, 814, 216, 40, 6, 1;
%e 0, 8390, 16220, 8057, 2171, 400, 57, 7, 1;
%e 0, 86796, 182714, 93627, 25628, 4740, 666, 77, 8, 1; ...
%e This triangle equals matrix product P^-1*R,
%e which equals triangle P shifted right one column,
%e where P = A135880 begins:
%e 1;
%e 1, 1;
%e 2, 2, 1;
%e 6, 7, 3, 1;
%e 25, 34, 15, 4, 1;
%e 138, 215, 99, 26, 5, 1;
%e 970, 1698, 814, 216, 40, 6, 1; ...
%e and Q = P^2 = A135885 begins:
%e 1;
%e 2, 1;
%e 6, 4, 1;
%e 25, 20, 6, 1;
%e 138, 126, 42, 8, 1;
%e 970, 980, 351, 72, 10, 1;
%e 8390, 9186, 3470, 748, 110, 12, 1; ...
%e and R = A135894 begins:
%e 1;
%e 1, 1;
%e 2, 3, 1;
%e 6, 12, 5, 1;
%e 25, 63, 30, 7, 1;
%e 138, 421, 220, 56, 9, 1;
%e 970, 3472, 1945, 525, 90, 11, 1; ...
%e where column k of R equals column 0 of P^(2k+1),
%e and column k of Q=P^2 equals column 0 of P^(2k+2), for k>=0.
%o (PARI) {T(n,k)=local(P=Mat(1),R=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]))));R=P*PShR; R=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,R[r,c], if(c==1,(P^2)[ #P,1],(P^(2*c-1))[r-c+1,1])))); P=matrix(#R, #R, r,c, if(r>=c, if(r<#R,P[r,c], (R^c)[r-c+1,1])))));(P^-1*R)[n+1,k+1]}
%Y Cf. A135880 (P), A135885 (Q=P^2), A135894 (R); A135899 (P*R^-1*P), A135900 (R^-1*Q).
%K nonn,tabl
%O 0,8
%A _Paul D. Hanna_, Dec 15 2007
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