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 A117269 Triangle T, read by rows, that satisfies matrix equation: T - (T-I)^2 = C, where C is Pascal's triangle. 3

%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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Last modified December 7 05:59 EST 2023. Contains 367630 sequences. (Running on oeis4.)