A324123 Irregular triangle read by rows: row n gives numerators of coefficients of polynomials arising from Chebyshev quadrature.

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

Table of n, a(n) for n=0..48.

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

Adjacent sequences: A324120 A324121 A324122 * A324124 A324125 A324126

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