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A136672
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Riordan array ((1+x^2)/(1-x)^2, -x/(1-x)^2).
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0
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1, 2, -1, 4, -4, 1, 6, -11, 6, -1, 8, -24, 22, -8, 1, 10, -45, 62, -37, 10, -1, 12, -76, 147, -128, 56, -12, 1, 14, -119, 308, -366, 230, -79, 14, -1, 16, -176, 588, -912, 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
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OFFSET
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1,2
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COMMENTS
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Row sums are: {1, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, ...}
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LINKS
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FORMULA
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G.f.: g(x,t)=(2 - x)/(1 - (2 - x)*t + t^2).
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EXAMPLE
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{1},
{2, -1},
{4, -4, 1},
{6, -11, 6, -1},
{8, -24, 22, -8, 1},
{10, -45, 62, -37, 10, -1},
{12, -76, 147, -128, 56, -12, 1},
{14, -119,308, -366, 230, -79, 14, -1},
{16, -176, 588, -912, 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}
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MATHEMATICA
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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}]
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]
{1} ~ Join ~ CoefficientList[Table[(2 - x) ChebyshevU[n - 2, 1 - x/2], {n, 2, 10}], x] // Flatten (* Eric W. Weisstein, May 10 2017 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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