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%I #67 Mar 27 2023 16:55:34
%S 1,1,3,-1,2,-1,45,-30,1,72,-60,7,105,-105,21,-1,6480,-7560,2142,-149,
%T 42525,-56700,20790,-2220,-43,22400,-33600,15120,-2280,53,56133,
%U -93555,49896,-9900,561,-43,32659200,-59875200,36923040,-9163440,817674,-33889,7882875,-15765750,11036025,-3303300,405405,-19110,-2161
%N Irregular triangle read by rows: row n gives numerators of coefficients of polynomials arising from Chebyshev quadrature.
%C High-order coefficients first. The polynomials have been normalized.
%C From _Petros Hadjicostas_, Oct 28 2019: (Start)
%C Row n >= 0 of this array corresponds to the polynomial Sum_{k = 0..m} T(2*m, k)*x^(2*(m-k))/A002680(2*m) when k = 2*m and to the polynomial Sum_{k = 0..m} T(2*m+1, k)*x^(2*(m-k)+1)/A002680(2*m+1) when n = 2*m+1.
%C The same numbers appear in array A101270 but with zeros for the missing powers and with the order of the powers reversed in each row (from lower-order powers to higher-order powers).
%C For Maple programs to generate the rows of this array, see the link and the program section.
%C (End)
%H Petros Hadjicostas, <a href="/A324123/a324123.txt">Alternative Maple program</a>.
%H H. E. Salzer, <a href="https://doi.org/10.1002/sapm1947261191">Tables for facilitating the use of Chebyshev's quadrature formula</a>, Journal of Mathematics and Physics, 26 (1947), 191-194.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChebyshevQuadrature.html">Chebyshev Quadrature</a>.
%e Triangle begins:
%e 1;
%e 1;
%e 3, -1;
%e 2, -1,
%e 45, -30, 1;
%e 72, -60, 7,
%e 105, -105, 21, -1;
%e 6480, -7560, 2142, -149;
%e 42525, -56700, 20790, -2220, -43;
%e ...
%p gf := n -> exp(n*(arctanh(z)/z + 1/2*log(-z^2 + 1) - 1)):
%p ser := n -> series(gf(n), z, n + 2):
%p g := n -> ilcm(seq(denom(coeff(ser(n), z, k)), k = 0..n)):
%p a := proc(n) local G, S; G:=g(n); S:=ser(n); seq(G*coeff(S,z,m), m=0..n,2) end:
%p seq(a(n), n=0..12); # _Petros Hadjicostas_, Oct 28 2019
%t row[0] = row[1] = {1}; row[n_] := row[n] = Select[Normal[z^n* Exp[-n*HypergeometricPFQ[{1, 1, 3/2}, {2, 5/2}, 1/z^2]/(6 z^2)] + O[z, Infinity]^n], PolynomialQ[#, z]&] // Together // Numerator // CoefficientList[#, z]& // Reverse // DeleteCases[#, 0]&;
%t T[n_, k_] := row[n][[k + 1]];
%t Table[T[n, k], {n, 0, 12}, {k, 0, If[n == 1, 0, Length[row[n]] - 1]}] // Flatten (* _Jean-François Alcover_, Mar 26 2023 *)
%Y For denominators see A002680 (which is also the first column).
%Y Cf. A101270 (aerated version of this array in reverse order).
%K sign,tabf,frac
%O 0,3
%A _N. J. A. Sloane_, Feb 15 2019
%E More terms from _Petros Hadjicostas_, Oct 28 2019
%E T(0,0) = 1 prepended by _Petros Hadjicostas_, Oct 28 2019
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