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A324123
Irregular triangle read by rows: row n gives numerators of coefficients of polynomials arising from Chebyshev quadrature.
3
1, 1, 3, -1, 2, -1, 45, -30, 1, 72, -60, 7, 105, -105, 21, -1, 6480, -7560, 2142, -149, 42525, -56700, 20790, -2220, -43, 22400, -33600, 15120, -2280, 53, 56133, -93555, 49896, -9900, 561, -43, 32659200, -59875200, 36923040, -9163440, 817674, -33889, 7882875, -15765750, 11036025, -3303300, 405405, -19110, -2161
OFFSET
0,3
COMMENTS
High-order coefficients first. The polynomials have been normalized.
From Petros Hadjicostas, Oct 28 2019: (Start)
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.
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).
For Maple programs to generate the rows of this array, see the link and the program section.
(End)
LINKS
Petros Hadjicostas, Alternative Maple program.
H. E. Salzer, Tables for facilitating the use of Chebyshev's quadrature formula, Journal of Mathematics and Physics, 26 (1947), 191-194.
Eric Weisstein's World of Mathematics, Chebyshev Quadrature.
EXAMPLE
Triangle begins:
1;
1;
3, -1;
2, -1,
45, -30, 1;
72, -60, 7,
105, -105, 21, -1;
6480, -7560, 2142, -149;
42525, -56700, 20790, -2220, -43;
...
MAPLE
gf := n -> exp(n*(arctanh(z)/z + 1/2*log(-z^2 + 1) - 1)):
ser := n -> series(gf(n), z, n + 2):
g := n -> ilcm(seq(denom(coeff(ser(n), z, k)), k = 0..n)):
a := proc(n) local G, S; G:=g(n); S:=ser(n); seq(G*coeff(S, z, m), m=0..n, 2) end:
seq(a(n), n=0..12); # Petros Hadjicostas, Oct 28 2019
MATHEMATICA
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[n_, k_] := row[n][[k + 1]];
Table[T[n, k], {n, 0, 12}, {k, 0, If[n == 1, 0, Length[row[n]] - 1]}] // Flatten (* Jean-François Alcover, Mar 26 2023 *)
CROSSREFS
For denominators see A002680 (which is also the first column).
Cf. A101270 (aerated version of this array in reverse order).
Sequence in context: A364906 A188584 A103514 * A016570 A070773 A046804
KEYWORD
sign,tabf,frac
AUTHOR
N. J. A. Sloane, Feb 15 2019
EXTENSIONS
More terms from Petros Hadjicostas, Oct 28 2019
T(0,0) = 1 prepended by Petros Hadjicostas, Oct 28 2019
STATUS
approved