A204180 - OEIS

%I #18 Oct 24 2024 05:40:53

%S 1,-1,1,-3,1,1,-9,6,-1,-2,-32,32,-10,1,-34,-132,183,-81,15,-1,-324,

%T -604,1159,-655,170,-21,1,-2988,-2860,8137,-5589,1825,-316,28,-1,

%U -28944,-11864,62852,-51184,19894,-4326,539,-36,1,-300816,-8568

%N Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (f(i,j)), where f(i,1)=f(1,j)=1, f(i,i)= i; f(i,j)=0 otherwise; as in A204179.

%C Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are real, and they interlace the zeros of p(n+1). See A202605 and A204016 for guides to related sequences.

%D (For references regarding interlacing roots, see A202605.)

%H Robert Israel, <a href="/A204180/b204180.txt">Table of n, a(n) for n = 1..10010</a> (rows 1 to 140, flattened)

%F From _Robert Israel_, Jun 26 2018: (Start)

%F p(n,x) = (1 - Sum_{k=2..n} 1/((1-x)*(k-x)))*Product_{k=1..n} (k - x).

%F p(n+1,x) = (n+1-x)*p(n,x) - Gamma(n+1-x)/Gamma(2-x). (End)

%e Top of the array:

%e    1,  -1;

%e    1,  -3,   1;

%e    1,  -9,   6,  -1;

%e   -2, -32,  32, -10,   1;

%p f:= proc(n) local P;

%p   P:= normal(mul(i-lambda,i=1..n)*(1 - add(1/(lambda-1)/(lambda-i),i=2..n)));

%p   seq(coeff(P,lambda,i),i=0..n);

%p end proc:

%p seq(f(n),n=1..20); # _Robert Israel_, Jun 26 2018

%t f[i_, j_] := 0; f[1, j_] := 1; f[i_, 1] := 1; f[i_, i_] := i;

%t m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]

%t TableForm[m[8]] (* 8x8 principal submatrix *)

%t Flatten[Table[f[i, n + 1 - i],

%t   {n, 1, 15}, {i, 1, n}]]  (* A204179 *)

%t p[n_] := CharacteristicPolynomial[m[n], x];

%t c[n_] := CoefficientList[p[n], x]

%t TableForm[Flatten[Table[p[n], {n, 1, 10}]]]

%t Table[c[n], {n, 1, 12}]

%t Flatten[%]                 (* A204180 *)

%t TableForm[Table[c[n], {n, 1, 10}]]

%Y Cf. A204179, A202605, A204016.

%K tabf,sign

%O 1,4

%A _Clark Kimberling_, Jan 12 2012