OFFSET
1,4
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.
EXAMPLE
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).
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 *)
PROG
(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)))))))
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Jan 28 2005
STATUS
approved