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A053122 Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).

%I A053122 #104 Feb 16 2025 08:32:42
%S A053122 1,-2,1,3,-4,1,-4,10,-6,1,5,-20,21,-8,1,-6,35,-56,36,-10,1,7,-56,126,
%T A053122 -120,55,-12,1,-8,84,-252,330,-220,78,-14,1,9,-120,462,-792,715,-364,
%U A053122 105,-16,1,-10,165,-792,1716,-2002,1365,-560,136,-18,1,11,-220,1287,-3432,5005,-4368,2380,-816,171,-20
%N A053122 Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).
%C A053122 Apart from signs, identical to A078812.
%C A053122 Another version with row-leading 0's and differing signs is given by A285072.
%C A053122 G.f. for row polynomials S(n,x-2) (signed triangle): 1/(1+(2-x)*z+z^2). Unsigned triangle |a(n,m)| has g.f. 1/(1-(2+x)*z+z^2) for row polynomials.
%C A053122 Row sums (signed triangle) A049347(n) (periodic(1,-1,0)). Row sums (unsigned triangle) A001906(n+1)=F(2*(n+1)) (even-indexed Fibonacci).
%C A053122 In the language of Shapiro et al. (see A053121 for the reference) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group.
%C A053122 The (unsigned) column sequences are A000027, A000292, A000389, A000580, A000582, A001288 for m=0..5, resp. For m=6..23 they are A010966..(+2)..A011000 and for m=24..49 they are A017713..(+2)..A017763.
%C A053122 Riordan array (1/(1+x)^2,x/(1+x)^2). Inverse array is A039598. Diagonal sums have g.f. 1/(1+x^2). - _Paul Barry_, Mar 17 2005. Corrected by _Wolfdieter_ Lang, Nov 13 2012.
%C A053122 Unsigned version is in A078812. - _Philippe Deléham_, Nov 05 2006
%C A053122 Also row n gives (except for an overall sign) coefficients of characteristic polynomial of the Cartan matrix for the root system A_n. - _Roger L. Bagula_, May 23 2007
%C A053122 From _Wolfdieter Lang_, Nov 13 2012: (Start)
%C A053122 The A-sequence for this Riordan triangle is A115141, and the Z-sequence is A115141(n+1), n>=0. For A- and Z-sequences for Riordan matrices see the W. Lang link under A006232 with details and references.
%C A053122 S(n,x^2-2) = sum(r(j,x^2),j=0..n) with Chebyshev's S-polynomials and r(j,x^2) := R(2*j+1,x)/x, where R(n,x) are the monic integer Chebyshv T-polynomials with coefficients given in A127672. Proof from comparing the o.g.f. of the partial sum of the r(j,x^2) polynomials (see a comment on the signed Riordan triangle A111125) with the present Riordan type o.g.f. for the row polynomials with x -> x^2.  (End)
%C A053122 S(n,x^2-2) = S(2*n+1,x)/x, n >= 0, from the odd part of the bisection of the o.g.f. - _Wolfdieter Lang_, Dec 17 2012
%C A053122 For a relation to a generator for the Narayana numbers A001263, see A119900, whose columns are unsigned shifted rows (or antidiagonals) of this array, referring to the tables in the example sections. - _Tom Copeland_, Oct 29 2014
%C A053122 The unsigned rows of this array are alternating rows of a mirrored A011973 and alternating shifted rows of A030528 for the Fibonacci polynomials. - _Tom Copeland_, Nov 04 2014
%C A053122 Boas-Buck type recurrence for column k >= 0 (see Aug 10 2017 comment in A046521 with references): a(n, m) =  (2*(m + 1)/(n - m))*Sum_{k = m..n-1} (-1)^(n-k)*a(k, m), with input a(n, n) = 1, and a(n,k) = 0 for n < k. - _Wolfdieter Lang_, Jun 03 2020
%C A053122 Row n gives the characteristic polynomial of the (n X n)-matrix M where M[i,j] = 2 if i = j, -1 if |i-j| = 1 and 0 otherwise. The matrix M is positive definite and has 2-condition number (cot(Pi/(2*n+2)))^2. - _Jianing Song_, Jun 21 2022
%C A053122 Also the convolution triangle of (-1)^(n+1)*n. - _Peter Luschny_, Oct 07 2022
%D A053122 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
%D A053122 Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
%D A053122 R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 62.
%D A053122 Sigurdur Helgasson, Differential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics, volume 34. A. M. S.: ISBN 0-8218-2848-7, 1978, p. 463.
%H A053122 T. D. Noe, <a href="/A053122/b053122.txt">Rows n=0..50 of triangle, flattened</a>
%H A053122 Wolfdieter Lang, <a href="/A053122/a053122.pdf">First rows of the triangle.</a>
%H A053122 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A053122 P. Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry4/barry64.html">Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays</a>, JIS 12 (2009) 09.8.6.
%H A053122 Naiomi T. Cameron and Asamoah Nkwanta, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Cameron/cameron46.html">On some (pseudo) involutions in the Riordan group</a>, J. of Integer Sequences, 8(2005), 1-16.
%H A053122 P. Damianou, <a href="http://arxiv.org/abs/1110.6620">On the characteristic polynomials of Cartan matrices and Chebyshev polynomials</a>, arXiv preprint arXiv:1110.6620 [math.RT], 2014 (p. 10). - _Tom Copeland_, Oct 11 2014
%H A053122 Pentti Haukkanen, Jorma Merikoski, Seppo Mustonen, <a href="http://www.acta.sapientia.ro/acta-math/C6-2/math62-5.pdf">Some polynomials associated with regular polygons</a>, Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 178-193.
%H A053122 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CartanMatrix.html">Cartan Matrix</a>
%H A053122 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DynkinDiagram.html">Dynkin Diagram</a>
%H A053122 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A053122 a(n, m) := 0 if n<m else ((-1)^(n-m))*binomial(n+m+1, 2*m+1);
%F A053122 a(n, m) = -2*a(n-1, m) + a(n-1, m-1) - a(n-2, m), a(n, -1) := 0 =: a(-1, m), a(0, 0)=1, a(n, m) := 0 if n<m;
%F A053122 O.g.f. for m-th column (signed triangle): ((x/(1+x)^2)^m)/(1+x)^2.
%F A053122 From _Jianing Song_, Jun 21 2022: (Start)
%F A053122 T(n,k) = [x^k]f_n(x), where f_{-1}(x) = 0, f_0(x) = 1, f_n(x) = (x-2)*f_{n-1}(x) - f_{n-2}(x) for n >= 2.
%F A053122 f_n(x) = (((x-2+sqrt(x^2-4*x))/2)^(n+1) - ((x-2-sqrt(x^2-4*x))/2)^(n+1))/sqrt(x^2-4x).
%F A053122 The roots of f_n(x) are 2 + 2*cos(k*Pi/(n+1)) = 4*(cos(k*Pi/(2*n+2)))^2 for 1 <= k <= n. (End)
%e A053122 The triangle a(n,m) begins:
%e A053122 n\m   0    1    2     3     4     5     6    7    8  9
%e A053122 0:    1
%e A053122 1:   -2    1
%e A053122 2:    3   -4    1
%e A053122 3:   -4   10   -6     1
%e A053122 4:    5  -20   21    -8     1
%e A053122 5:   -6   35  -56    36   -10     1
%e A053122 6:    7  -56  126  -120    55   -12     1
%e A053122 7:   -8   84 -252   330  -220    78   -14    1
%e A053122 8:    9 -120  462  -792   715  -364   105  -16    1
%e A053122 9:  -10  165 -792  1716 -2002  1365  -560  136  -18  1
%e A053122 ... Reformatted and extended by _Wolfdieter Lang_, Nov 13 2012
%e A053122 E.g., fourth row (n=3) {-4,10,-6,1} corresponds to the polynomial S(3,x-2) = -4+10*x-6*x^2+x^3.
%e A053122 From _Wolfdieter Lang_, Nov 13 2012: (Start)
%e A053122 Recurrence: a(5,1) = 35 = 1*5 + (-2)*(-20) -1*(10).
%e A053122 Recurrence from Z-sequence [-2,-1,-2,-5,...]: a(5,0) = -6 = (-2)*5 + (-1)*(-20) + (-2)*21 + (-5)*(-8) + (-14)*1.
%e A053122 Recurrence from A-sequence [1,-2,-1,-2,-5,...]: a(5,1) = 35 = 1*5  + (-2)*(-20) + (-1)*21 + (-2)*(-8) + (-5)*1.
%e A053122 (End)
%e A053122 E.g., the fourth row (n=3) {-4,10,-6,1} corresponds also to the polynomial S(7,x)/x = -4 + 10*x^2 - 6*x^4 + x^6. - _Wolfdieter Lang_, Dec 17 2012
%e A053122 Boas-Buck type recurrence: -56 = a(5, 2) = 2*(-1*1 + 1*(-6) - 1*21) = -2*28 = -56. - _Wolfdieter Lang_, Jun 03 2020
%p A053122 seq(seq((-1)^(n+m)*binomial(n+m+1,2*m+1),m=0..n),n=0..10); # _Robert Israel_, Oct 15 2014
%p A053122 # Uses function PMatrix from A357368. Adds a row above and a column to the left.
%p A053122 PMatrix(10, n -> -(-1)^n*n); # _Peter Luschny_, Oct 07 2022
%t A053122 T[n_, m_, d_] := If[ n == m, 2, If[n == m - 1 || n == m + 1, -1, 0]]; M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]; a = Join[M[1], Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]; Flatten[a] (* _Roger L. Bagula_, May 23 2007 *)
%t A053122 (* Alternative code for the matrices from MathWorld: *)
%t A053122 sln[n_] := 2IdentityMatrix[n] - PadLeft[PadRight[IdentityMatrix[n - 1], {n, n - 1}], {n, n}] - PadLeft[PadRight[IdentityMatrix[n - 1], {n - 1, n}], {n, n}] (* _Roger L. Bagula_, May 23 2007 *)
%o A053122 (Sage)
%o A053122 @CachedFunction
%o A053122 def A053122(n,k):
%o A053122     if n< 0: return 0
%o A053122     if n==0: return 1 if k == 0 else 0
%o A053122     return A053122(n-1,k-1)-A053122(n-2,k)-2*A053122(n-1,k)
%o A053122 for n in (0..9): [A053122(n,k) for k in (0..n)] # _Peter Luschny_, Nov 20 2012
%Y A053122 Cf. A005248, A127677, A053123, A049310.
%Y A053122 Cf. A011973, A030528, A034867, A119900.
%Y A053122 Cf. A285072 (version with row-leading 0's and differing signs). - _Eric W. Weisstein_, Apr 09 2017
%K A053122 easy,nice,sign,tabl
%O A053122 0,2
%A A053122 _Wolfdieter Lang_