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A102610 Triangle read by rows: coefficients of characteristic polynomials of lower triangular matrix of Robbins triangle numbers.

%I A102610 #17 Feb 21 2025 16:53:52
%S A102610 0,1,-1,1,-2,1,1,-4,5,-2,1,-11,33,-37,14,1,-53,495,-1423,1568,-588,1,
%T A102610 -482,23232,-213778,612035,-673260,252252,1,-7918,3607384,-172966930,
%U A102610 1590265243,-4551765520,5006613612,-1875745872,1,-226266,1732486848,-787838048562,37768573496883,-347235787044084
%N A102610 Triangle read by rows: coefficients of characteristic polynomials of lower triangular matrix of Robbins triangle numbers.
%C A102610 Roots of n-th characteristic polynomial are the first n Robbins numbers (A005130).
%C A102610 Second column of triangle is partial sums of Robbins numbers negated (A173312).
%e A102610 Generation of the triangle:
%e A102610 We begin with A048601
%e A102610 1
%e A102610 1 1
%e A102610 2 3 2
%e A102610 7 14 14 7
%e A102610 42 105 135 105 42
%e A102610 ...
%e A102610 and get polynomials
%e A102610 x - 1
%e A102610 x^2 - 2*x + 1
%e A102610 x^3 - 4*x^2 + 5*x - 2
%e A102610 x^4 - 11*x^3 + 33*x^2 - 37*x + 14
%e A102610 x^5 - 53*x^4 + 495*x^3 - 1423*x^2 + 1568*x - 588
%e A102610 ...
%o A102610 (PARI)
%o A102610 T(n, k) = binomial(n+k-2,k-1)*((2*n-k-1)!/(n-k)!)*prod(j=0,n-2,((3*j+1)!/(n+j)!))
%o A102610 RM(n)=M=matrix(n,n);for(l=1,n, for(k=1,l,M[l,k]=T(l,k)));M
%o A102610 for(i=1,10,print(charpoly(RM(i))))
%Y A102610 Cf. A005130, A048601.
%K A102610 sign,tabl
%O A102610 0,5
%A A102610 Lambert Klasen (lambert.klasen(AT)gmx.net) and _Gary W. Adamson_, Jan 30 2005
%E A102610 Sequence has been prepended with a(0)=0 to enable table display (so offset has been set to 0 accordingly) by _Michel Marcus_, Aug 23 2013