A002680 Denominators of coefficients of polynomials arising from Chebyshev quadrature. (Formerly M2261 N0892)

1, 1, 3, 2, 45, 72, 105, 6480, 42525, 22400, 56133, 32659200, 7882875, 452656512000, 351833625, 129153024, 4396161144375, 8473729904640000, 9901766875, 54452188367216640000, 4167666272734965, 168568418304000000, 7426884628582453125, 20484913263746899968000000

Offset

0,3

References

C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 184.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Petros Hadjicostas, Table of n, a(n) for n = 0..30

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.

Index entries for sequences related to Chebyshev polynomials.

Maple

s := proc(z) 1/6*z^2*hypergeom([1, 1, 3/2], [2, 5/2], z^2); end proc;
f:= proc(n) local v, m; v := 1;
     for m to n do
     v := lcm(v, denom(coeftayl(exp(-n*s(z)), z = 0, m))); end do; v;
     end proc;
seq(f(n), n = 0 .. 26); # Petros Hadjicostas, Oct 28 2019

Mathematica

gf[n_, z_] := Exp[n (ArcTanh[z]/z + Log[1 - z^2]/2 - 1)];
ser[n_] := CoefficientList[Series[gf[n, z], {z, 0, n}], z];
Table[LCM @@ Denominator[ser[n]], {n, 0, 30}] (* Peter Luschny, Oct 28 2019 *)

Crossrefs

Numerators appear in A101270 (with zeros for the missing powers) and in A324123 (without the zeros for the missing powers and with the highest powers first).

Sequence in context: A065085 A321169 A093398 * A009395 A339513 A375252

Adjacent sequences: A002677 A002678 A002679 * A002681 A002682 A002683

Keyword

nonn,easy,frac

Author

N. J. A. Sloane

Extensions

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 11 2004

a(22)-a(30) from Petros Hadjicostas, Oct 28 2019

a(0) = 1 prepended by Petros Hadjicostas, Oct 28 2019

Status

approved