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%I #7 Sep 13 2012 10:04:12
%S 1,-1,1,-3,2,1,-8,17,-10,1,-23,137,-265,150,1,-75,1333,-7389,13930,
%T -7800,1,-278,16558,-277988,1513897,-2835590,1583400,1,-1155,260364,
%U -14799354,245309373,-1330523259,2488395830,-1388641800,1,-5295,5042064,-1092706314,61514634933,-1016911327479
%N Triangle read by rows: Characteristic polynomials of lower triangular Bell number matrix.
%C Roots of the polynomials are the Bell numbers (A000110) except the leading term.
%C Second column of the triangle = A024716(n) (partial sums of Bell numbers).
%C 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.
%C So from triangle
%C 1
%C 1 2
%C 2 3 5
%C 5 7 10 15
%C ...
%C we get characteristic polynomials
%C x - 1
%C x^2 - 3*x + 2
%C x^3 - 8*x^2 + 17*x - 10
%C x^4 - 23*x^3 + 137*x^2 - 265*x + 150
%C ...
%C All polynomials (except the first) evaluated at 2 give zero.
%e The characteristic polynomial of the 3X3 matrix
%e 1 0 0
%e 1 2 0
%e 2 3 5
%e = x^3 - 8x^2 + 17x - 10, with roots (1, 2, 5).
%t 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 *)
%o (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)))))))
%Y Cf. A000110, A024716.
%K sign,tabl
%O 1,4
%A Lambert Klasen (lambert.klasen(AT)gmx.net) and _Gary W. Adamson_, Jan 28 2005
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