A135895 - OEIS

%I #2 Mar 30 2012 18:37:08

%S 1,2,1,7,6,1,34,39,10,1,215,300,95,14,1,1698,2741,990,175,18,1,16220,

%T 29380,11635,2296,279,22,1,182714,363922,154450,32865,4410,407,26,1,

%U 2378780,5135894,2302142,517916,74319,7524,559,30,1,35219202,81557270

%N Triangle, read by rows, equal to R^2, the matrix square of R = A135894.

%C Triangle P = A135880 is defined by: column k of P^2 equals column 0 of P^(2k+2) such that column 0 of P^2 equals column 0 of P shift left.

%F Column k of R^2 = column 1 of P^(2k+1) for k>=0 where triangle P = A135880; column 0 of R^2 = column 1 of P; column 1 of R^2 = column 1 of P^3; column 2 of R^2 = column 1 of P^5.

%e Triangle R^2 begins:

%e 1;

%e 2, 1;

%e 7, 6, 1;

%e 34, 39, 10, 1;

%e 215, 300, 95, 14, 1;

%e 1698, 2741, 990, 175, 18, 1;

%e 16220, 29380, 11635, 2296, 279, 22, 1;

%e 182714, 363922, 154450, 32865, 4410, 407, 26, 1;

%e 2378780, 5135894, 2302142, 517916, 74319, 7524, 559, 30, 1;

%e 35219202, 81557270, 38229214, 8980944, 1353522, 145805, 11830, 735, 34, 1;

%e where 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 = column 0 of P^(2k+1)

%e and 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 where column k of P equals column 0 of R^(k+1).

%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])))));(R^2)[n+1,k+1]}

%Y Cf. A135882 (column 0), A135890 (column 1); A135894 (R), A135880 (P), A135888 (P^3), A135892 (P^5).

%K nonn,tabl

%O 0,2

%A _Paul D. Hanna_, Dec 15 2007