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%I #5 Mar 30 2012 17:25:10
%S 1,-1,1,-3,2,1,-8,17,-10,1,-22,129,-248,140,1,-64,1053,-5666,10556,
%T -5880,1,-196,9501,-144662,758468,-1399272,776160,1,-625,93585,
%U -4220591,62818466,-326782044,601063848,-332972640,1,-2055,987335,-138047141,6098263596,-90157188424,467899386768
%N Triangle read by rows: Coefficients of characteristic polynomials of lower triangular matrix of Catalan numbers.
%C 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.)
%e Generation of the triangle:
%e We begin with A050166 (triangle)
%e 1
%e 1 2
%e 1 4 5
%e 1 6 14 14
%e 1 8 27 48 42
%e ...
%e and get polynomials
%e x - 1
%e x^2 - 3*x + 2
%e x^3 - 8*x^2 + 17*x - 10
%e x^4 - 22*x^3 + 129*x^2 - 248*x + 140
%e x^5 - 64*x^4 + 1053*x^3 - 5666*x^2 + 10556*x - 5880
%e ...
%o (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)))))))
%Y Cf. A000108, A014138.
%K sign,tabl
%O 1,4
%A Lambert Klasen (lambert.klasen(AT)gmx.net) and _Gary W. Adamson_, Jan 29 2005
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