A111125 -id:A111125 - cp's OEIS frontend

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

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

Offset: 1

Roger L. Bagula, Apr 04 2008, Apr 12 2008

Row sums are: {1, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, ...}

			{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}

Pentti Haukkanen, Jorma Merikoski, Seppo Mustonen, Some polynomials associated with regular polygons, Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 178-193.

Cf. A053122, A111125.

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 *)

G.f.: g(x,t)=(2 - x)/(1 - (2 - x)*t + t^2).

Edited by Ralf Stephan, Feb 07 2014

A220669 Coefficient array for powers of x^2 of the square of Chebyshev's C(2*n+1,x)/x =: tau(n,x) polynomials.

Original entry on oeis.org

1, 9, -6, 1, 25, -50, 35, -10, 1, 49, -196, 294, -210, 77, -14, 1, 81, -540, 1386, -1782, 1287, -546, 135, -18, 1, 121, -1210, 4719, -9438, 11011, -8008, 3740, -1122, 209, -22, 1, 169, -2366, 13013, -37180, 63206, -68952, 50388, -25194, 8645, -2002, 299, -26, 1

Offset: 0

Wolfdieter Lang, Jan 04 2013

The row lengths sequence of this irregular triangle is 2*n+1 = A005408(n).
For the coefficient array of powers of x^2 of the monic integer Chebyshev C(2*n+1,x)/x = :tau(n,x) polynomials see the signed triangle ((-1)^(n-m))*A111125(n,m). See the comment from Oct 23 2012.
The o.g.f. of the row polynomials p(n,x) := sum(a(n,m)*x^m, m=0..2*n), n>=0, is G2(x,z) = sum(p(n,x)*z^n,n=0..infinity) =  (1+ (6-2*x)*z+z^2)/((1-z)*((z+1)^2-z*(x-2)^2)). Derived from the odd part of the bisection of the o.g.f. for the C(n,x)^2 polynomials. Note that p(n,x) = (tau(n,sqrt(x)))^2.

			The array begins:
n\m  0     1    2     3     4     5    6     7    8   9  10
0:   1
1:   9    -6    1
2:  25   -50   35   -10     1
3:  49  -196  294  -210    77   -14    1
4:  81  -540 1386 -1782  1287  -546  135   -18    1
5: 121 -1210 4719 -9438 11011 -8008 3740 -1122  209 -22   1
...
Row polynomial for n=1: p(1,x) = (tau(1,sqrt(x)))^2 = (3-1*x)^2 = 9 - 6*x +1*x^2.
Row polynomial for n=2: p(2,x) = (tau(2,sqrt(x)))^2 = (5 - 5*x + 1*x^2)^2 = 25 - 50*x + 35*x^2 - 10*x^3 + 1*x^4.

Odd rows of A156308 with alternating signs of elements.

Cf. A005408, A111125.

a(n,m) = [x^m] (p(n,x)), n>=0, 0 <= m <= 2*n, with p(n,x) = (C(2*n+1,sqrt(x))/sqrt(x))^2 = (tau(n,sqrt(x)))^2, For Chebyshev's C and tau polynomials see a comment above.
For n >= 0, 0 <= m <= 2*n, a(n,m) = [x^m*z^n] G2(x,z), where the o.g.f. G2(x,z) given in a comment above.
a(n,m) = (-1)^m * A156308(2*n+1,m+1). - Max Alekseyev, Mar 06 2018

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

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