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%I #3 Mar 30 2012 18:36:56
%S 1,1,1,3,2,1,19,9,3,1,207,76,18,4,1,3211,1035,190,30,5,1,64383,19266,
%T 3105,380,45,6,1,1581259,450681,67431,7245,665,63,7,1,45948927,
%U 12650072,1802724,179816,14490,1064,84,8,1,1541641771,413540343,56925324,5408172
%N Triangle T, read by rows, that satisfies matrix equation: T - (T-I)^2 = C, where C is Pascal's triangle.
%C E.g.f. of column 0 is F(x) = (3-sqrt(5-4*exp(x)))/2 since F(x) satisfies the characteristic equation: F - (F-1)^2 = exp(x). The matrix log of T is the integer triangle A117270.
%F T(n,k) = A052886(n-k)*C(n,k) for n>k, with T(n,n) = 1.
%e Triangle T begins:
%e 1;
%e 1,1;
%e 3,2,1;
%e 19,9,3,1;
%e 207,76,18,4,1;
%e 3211,1035,190,30,5,1;
%e 64383,19266,3105,380,45,6,1;
%e 1581259,450681,67431,7245,665,63,7,1; ...
%e where (T-I)^2 =
%e 0;
%e 0,0;
%e 2,0,0;
%e 18,6,0,0;
%e 206,72,12,0,0;
%e 3210,1030,180,20,0,0;
%e 64382,19260,3090,360,30,0,0; ...
%e and T - (T-I)^2 = Pascal's triangle.
%o (PARI) {T(n,k)=local(C=matrix(n+1,n+1,r,c,if(r>=c,binomial(r-1,c-1))),M=C); for(i=1,n+1,M=(M-M^0)^2+C);return(M[n+1,k+1])}
%Y Cf. A117270 (log), A117271, A052886.
%K nonn,tabl
%O 0,4
%A _Paul D. Hanna_, Mar 05 2006
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