A101413 Triangle read by rows: Coefficients of characteristic polynomials of lower triangular matrix of Catalan numbers.

1, -1, 1, -3, 2, 1, -8, 17, -10, 1, -22, 129, -248, 140, 1, -64, 1053, -5666, 10556, -5880, 1, -196, 9501, -144662, 758468, -1399272, 776160, 1, -625, 93585, -4220591, 62818466, -326782044, 601063848, -332972640, 1, -2055, 987335, -138047141, 6098263596, -90157188424, 467899386768

Offset: 1

Author

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Jan 29 2005

Keywords

• sign
• tabl

Comments

Roots of n-th characteristic polynomial are the first n Catalan numbers (A000108). Second column of triangle is A014138(n) (Partial sums of Catalan numbers.)

Examples

			Generation of the triangle:
We begin with A050166 (triangle)
1
1 2
1 4 5
1 6 14 14
1 8 27 48 42
...
and get polynomials
x - 1
x^2 - 3*x + 2
x^3 - 8*x^2 + 17*x - 10
x^4 - 22*x^3 + 129*x^2 - 248*x + 140
x^5 - 64*x^4 + 1053*x^3 - 5666*x^2 + 10556*x - 5880
...
		

Crossrefs

Cf. A000108, A014138.

Programs

PARI

a(n,k)=binomial(2*n+1,k)*2*(n-k+1)/(2*n-k+2); CM(n)=M=matrix(n,n);for(l=0,n-1, for(k=0,l,M[l+1,k+1]=a(l,k)));M for(i=1,10,print(charpoly(CM(i)))) for(i=1,10,print(round(real(polroots(charpoly(CM(i)))))))