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A204021 Triangle read by rows: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min(2i-1,2j-1) (A157454).

%I A204021 #30 May 13 2025 12:08:33
%S A204021 1,1,-1,2,-4,1,4,-12,9,-1,8,-32,40,-16,1,16,-80,140,-100,25,-1,32,
%T A204021 -192,432,-448,210,-36,1,64,-448,1232,-1680,1176,-392,49,-1,128,-1024,
%U A204021 3328,-5632,5280,-2688,672,-64,1,256,-2304,8640,-17472,20592
%N A204021 Triangle read by rows: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min(2i-1,2j-1) (A157454).
%C A204021 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.
%C A204021 a(0)=1 by convention. - _Philippe Deléham_, Nov 17 2013
%C A204021 The n roots of the n-th polynomial are 1/(1+cos((2*k-1)*Pi/(2*n))) for k = 1..n. See my pdf in the link section for the proof. - _Jianing Song_, Dec 01 2023
%D A204021 (For references regarding interlacing roots, see A202605.)
%H A204021 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Morgan-VoycePolynomials.html">Morgan-Voyce polynomials</a>
%H A204021 Jianing Song, <a href="/A204021/a204021.pdf">Roots of the characteristic polynomials</a>
%F A204021 From _Peter Bala_, May 01 2012: (Start)
%F A204021 The triangle appears to be a signed version of the row reverse of A211957.
%F A204021 If true, then for 0 <= k <= n-1, T(n,k) = (-1)^k*n/(n-k)*2^(n-k-1)*binomial(2*n-k-1,k) and Sum_{k = 0..n} T(n,k)*x^(n-k) = 1/2*(-1)^n*(b(2*n,-2*x) + 1)/b(n,-2*x), where b(n,x) := Sum_{k = 0..n} binomial(n+k,2*k)*x^k are the Morgan-Voyce polynomials of A085478.
%F A204021 Conjectural o.g.f.: t*(1-x-x^2*t)/(1-2*t*(1-x)+t^2*x^2) = (1-x)*t + (2-4*x+x^2)*t^2 + .... (End)
%F A204021 T(n,k)=2*T(n-1,k)-2*T(n-1,k-1)-T(n-2,k-2), T(0,0)=T(1,0)=1, T(1,1)=-1, T(n,k)=0 of k<0 or if k>n. - _Philippe Deléham_, Nov 17 2013
%e A204021 Top of the triangle:
%e A204021   1
%e A204021   1....-1
%e A204021   2....-4.....1
%e A204021   4....-12....9....-1
%e A204021   8....-32....40...-16....1
%e A204021   16...-80....140..-100...25....-1
%e A204021   32...-192...432..-448...210...-36....1
%e A204021   ...
%e A204021 -448=2*(-100)-2*140-(-32). - _Philippe Deléham_, Nov 17 2013
%t A204021 f[i_, j_] := Min[2 i - 1, 2 j - 1];
%t A204021 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204021 TableForm[m[6]] (* 6x6 principal submatrix *)
%t A204021 Flatten[Table[f[i, n + 1 - i],
%t A204021   {n, 1, 15}, {i, 1, n}]]   (* A157454 *)
%t A204021 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204021 c[n_] := CoefficientList[p[n], x]
%t A204021 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204021 Table[c[n], {n, 1, 12}]
%t A204021 Flatten[%]                  (* A204021 *)
%t A204021 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204021 Cf. A157454, A202605, A204016. A085478, A211957.
%K A204021 tabl,sign
%O A204021 0,4
%A A204021 _Clark Kimberling_, Jan 11 2012