A101908 Triangle read by rows: Characteristic polynomials of lower triangular Bell number matrix.

1, -1, 1, -3, 2, 1, -8, 17, -10, 1, -23, 137, -265, 150, 1, -75, 1333, -7389, 13930, -7800, 1, -278, 16558, -277988, 1513897, -2835590, 1583400, 1, -1155, 260364, -14799354, 245309373, -1330523259, 2488395830, -1388641800, 1, -5295, 5042064, -1092706314, 61514634933, -1016911327479

Offset: 1

Author

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Jan 28 2005

Keywords

• sign
• tabl

Comments

Roots of the polynomials are the Bell numbers (A000110) except the leading term.

Second column of the triangle = A024716(n) (partial sums of Bell numbers).

Generation of the triangle: n-th row polynomials are the characteristic polynomial of the lower triangular matrix of the first n rows of the Bell triangle.

So from triangle

1

1 2

2 3 5

5 7 10 15

...

we get characteristic polynomials

x - 1

x^2 - 3*x + 2

x^3 - 8*x^2 + 17*x - 10

x^4 - 23*x^3 + 137*x^2 - 265*x + 150

...

All polynomials (except the first) evaluated at 2 give zero.

Examples

			The characteristic polynomial of the 3X3 matrix
1 0 0
1 2 0
2 3 5
= x^3 - 8x^2 + 17x - 10, with roots (1, 2, 5).
		

Crossrefs

Cf. A000110, A024716.

Programs

Mathematica

m[0, 0] = 1; m[n_, 0] := m[n, 0] = m[n-1, n-1]; m[n_, k_] := m[n, k] = m[n, k-1] + m[n-1, k-1]; m[n_, k_] /; k > n = 0; bm[n_] := Table[m[n0, k], {n0, 0, n}, {k, 0, n}]; row[n_] := (coes = Reverse[ CoefficientList[ CharacteristicPolynomial[ bm[n], x], x]]; Sign[coes[[1]]]*coes); Flatten[ Table[ row[n], {n, 0, 7}]] (* Jean-François Alcover, Sep 13 2012 *)

PARI

BM(n) = M=matrix(n,n);M[1,1]=1;if(n>1,M[2,1]=1;M[2,2]=2);\ for(l=3,n,M[l,1]=M[l-1,l-1];for(k=2,l,M[l,k]=M[l,k-1]+M[l-1,k-1]));M for(i=1,10,print(charpoly(BM(i)))) for(i=1,10,print(round(real(polroots(charpoly(BM(i)))))))