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A136672 Riordan array ((1+x^2)/(1-x)^2, -x/(1-x)^2).

%I A136672 #16 May 08 2025 08:56:10
%S A136672 1,2,-1,4,-4,1,6,-11,6,-1,8,-24,22,-8,1,10,-45,62,-37,10,-1,12,-76,
%T A136672 147,-128,56,-12,1,14,-119,308,-366,230,-79,14,-1,16,-176,588,-912,
%U A136672 770,-376,106,-16,1,18,-249,1044,-2046,2222,-1443,574,-137,18,-1,20,-340,1749,-4224,5720,-4732,2485,-832,172,-20,1
%N A136672 Riordan array ((1+x^2)/(1-x)^2, -x/(1-x)^2).
%C A136672 Row sums are: {1, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, ...}
%H A136672 Pentti Haukkanen, Jorma Merikoski, Seppo Mustonen, <a href="https://acta.sapientia.ro/content/docs/10415--Volume6_Issue_2_05_paper-201089.pdf">Some polynomials associated with regular polygons</a>, Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 178-193.
%F A136672 G.f.: g(x,t)=(2 - x)/(1 - (2 - x)*t + t^2).
%e A136672 {1},
%e A136672 {2, -1},
%e A136672 {4, -4, 1},
%e A136672 {6, -11, 6, -1},
%e A136672 {8, -24, 22, -8, 1},
%e A136672 {10, -45, 62, -37, 10, -1},
%e A136672 {12, -76, 147, -128, 56, -12, 1},
%e A136672 {14, -119,308, -366, 230, -79, 14, -1},
%e A136672 {16, -176, 588, -912, 770, -376, 106, -16, 1},
%e A136672 {18, -249, 1044, -2046, 2222, -1443, 574, -137, 18, -1},
%e A136672 {20, -340, 1749, -4224, 5720, -4732, 2485, -832, 172, -20, 1}
%t A136672 T[n_, m_, d_] := If[ n == m, 2, If[n == d && m == d - 1, 0, If[(n == m - 1 || n == m + 1), -1, 0]]]; M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]; Table[Det[M[d]], {d, 1, 10}]; Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}]; a = Join[{{1}}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]],x], {d, 1, 10}]]; Flatten[a] (* polynomial recursion*) Clear[p] p[x, 0] = 1; p[x, 1] = (2 - x); p[x, 2] = 4 - 4 x + x^2; p[x_, n_] := p[x, n] = (2 - x)*p[x, n - 1] - p[x, n - 2]; Table[ExpandAll[p[x, n]], {n, 0, Length[g] - 1}]
%t A136672 p[t_] = (2 - x)/(1 - (2 - x)*t + t^2); Table[ ExpandAll[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[ CoefficientList[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a]
%t A136672 {1} ~ Join ~ CoefficientList[Table[(2 - x) ChebyshevU[n - 2, 1 - x/2], {n, 2, 10}], x] // Flatten  (* _Eric W. Weisstein_, May 10 2017 *)
%Y A136672 Cf. A053122, A111125.
%K A136672 tabl,sign
%O A136672 1,2
%A A136672 _Roger L. Bagula_, Apr 04 2008, Apr 12 2008
%E A136672 Edited by _Ralf Stephan_, Feb 07 2014