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

1, 1, 1, 3, 2, 1, 19, 9, 3, 1, 207, 76, 18, 4, 1, 3211, 1035, 190, 30, 5, 1, 64383, 19266, 3105, 380, 45, 6, 1, 1581259, 450681, 67431, 7245, 665, 63, 7, 1, 45948927, 12650072, 1802724, 179816, 14490, 1064, 84, 8, 1, 1541641771, 413540343, 56925324, 5408172

Offset: 0

Author

Paul D. Hanna, Mar 05 2006

Keywords

• nonn
• tabl

Comments

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.

Examples

			Triangle T begins:
1;
1,1;
3,2,1;
19,9,3,1;
207,76,18,4,1;
3211,1035,190,30,5,1;
64383,19266,3105,380,45,6,1;
1581259,450681,67431,7245,665,63,7,1; ...
where (T-I)^2 =
0;
0,0;
2,0,0;
18,6,0,0;
206,72,12,0,0;
3210,1030,180,20,0,0;
64382,19260,3090,360,30,0,0; ...
and T - (T-I)^2 = Pascal's triangle.
		

Crossrefs

Cf. A117270 (log), A117271, A052886.

Programs

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])}

Formula

T(n,k) = A052886(n-k)*C(n,k) for n>k, with T(n,n) = 1.