cp's OEIS Frontend

A123588 Triangle read by rows: T(n, k) is the coefficient of x^k in the polynomial 1 - ChT(n, x^(1/2))^2, where ChT(n, x) is the n-th Chebyshev polynomial of the first kind, evaluated at x (0 <= k <= n).

%I A123588 #31 Aug 12 2024 14:27:32
%S A123588 0,1,-1,0,4,-4,1,-9,24,-16,0,16,-80,128,-64,1,-25,200,-560,640,-256,0,
%T A123588 36,-420,1792,-3456,3072,-1024,1,-49,784,-4704,13440,-19712,14336,
%U A123588 -4096,0,64,-1344,10752,-42240,90112,-106496,65536,-16384,1,-81,2160,-22176,114048,-329472,559104,-552960,294912
%N A123588 Triangle read by rows: T(n, k) is the coefficient of x^k in the polynomial 1 - ChT(n, x^(1/2))^2, where ChT(n, x) is the n-th Chebyshev polynomial of the first kind, evaluated at x (0 <= k <= n).
%D A123588 G. B. Shabat and I. A. Voevodskii, Drawing curves over number fields, The Grothendieck Festschift, vol. 3, Birkhäuser, 1990, pp. 199-227.
%H A123588 G. C. Greubel, <a href="/A123588/b123588.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A123588 Gareth Jones and David Singerman, <a href="https://doi.org/10.1112/blms/28.6.561">Belyi Functions, Hypermaps and Galois Groups</a>, Bull. London Math. Soc., 28 (1996), 561-590.
%H A123588 Yuri Matiyasevich, <a href="http://logic.pdmi.ras.ru/~yumat/Journal/Chebyshev/chebysh.htm">Generalized Chebyshev polynomials</a>.
%H A123588 G. B. Shabat and A. Zvonkin, <a href="https://www.labri.fr/perso/zvonkin/Research/shabzvon.pdf">Plane trees and algebraic numbers</a>, Contemporary Math., 1994, vol. 178, 233-275.
%F A123588 T(n, 0) = (n mod 2); T(n,k) = (-1)^(n + k - 1)*4^(k - 1)*(2*binomial(n + k, 2*k) - binomial(n + k - 1, 2*k - 1)) for k > 0. - _Detlef Meya_, Aug 11 2024
%e A123588 Polynomials:
%e A123588 0,
%e A123588 1 - x^2,
%e A123588 4 x^2 - 4 x^4,
%e A123588 1 - 9 x^2 + 24 x^4 - 16 x^6,
%e A123588 16 x^2 - 80 x^4 + 128 x^6 - 64 x^8,
%e A123588 1 - 25 x^2 + 200 x^4 - 560 x^6 + 640 x^8 - 256 x^10
%e A123588 Triangle starts:
%e A123588   0;
%e A123588   1,  -1;
%e A123588   0,   4,  -4;
%e A123588   1,  -9,  24,  -16;
%e A123588   0,  16, -80,  128, -64;
%e A123588   1, -25, 200, -560, 640, -256;
%p A123588 with(orthopoly): for n from 0 to 9 do seq(coeff(expand((1-T(n,sqrt(x))^2)),x,k), k=0..n) od; # yields sequence in triangular form
%t A123588 row[0] = {0}; row[n_] := CoefficientList[1 - ChebyshevT[n, x^(1/2)]^2, x]; Table[row[n], {n, 0, 9}] // Flatten (* _Jean-François Alcover_, Jan 29 2016 *)
%t A123588 T[n_,k_]:=If[k==0,Mod[n,2],(-1)^(n+k-1)*4^(k-1)*(2*Binomial[n+k,2*k]-Binomial[n+k-1,2*k-1])];Flatten[Table[T[n,k],{n,0,9},{k,0,n}]] (* _Detlef Meya_, Aug 11 2024 *)
%Y A123588 Cf. A000004 (row sums vanish), A114619 (alternating row sums).
%K A123588 sign,tabl
%O A123588 0,5
%A A123588 _Gary W. Adamson_ and _Roger L. Bagula_, Nov 12 2006
%E A123588 Edited by _N. J. A. Sloane_, Dec 03 2006