A182823 Coefficient array of orthogonal polynomials P(n,x)=(x-n)*P(n-1,x)-(n-1)^2*P(n-2,x), P(0,x)=1, P(1,x)=x-1.

1, -1, 1, 1, -3, 1, 1, 6, -6, 1, -13, 4, 21, -10, 1, 49, -129, -5, 55, -15, 1, 31, 723, -624, -85, 120, -21, 1, -1981, -386, 5271, -2009, -385, 231, -28, 1, 14329, -34320, -11978, 25508, -4809, -1204, 406, -36, 1, -2177, 347913, -263862, -112974, 93429, -8757, -3066, 666, -45, 1, -1138879, -701387, 3956751, -1200270, -657735, 278523, -10983, -6810, 1035, -55, 1

Offset

0,5

Comments

Inverse is A182822.

Links

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened)

Formula

Exponential Riordan array [exp(-arctan(sqrt(3)*x/(x+2))/sqrt(3))/sqrt(1+x+x^2), 2*arctan(sqrt(3)*x/(x+2))/sqrt(3)].

Example

Triangle begins
1,
-1, 1,
1, -3, 1,
1, 6, -6, 1,
-13, 4, 21, -10, 1,
49, -129, -5, 55, -15, 1,
31, 723, -624, -85, 120, -21, 1,
-1981, -386, 5271, -2009, -385, 231, -28, 1,
14329, -34320, -11978, 25508, -4809, -1204, 406, -36, 1

Maple

f := proc (n) option remember; normal((x-n)*procname(n-1)-(n-1)^2*procname(n-2)) end proc:
f(0):= 1: f(1):= x-1:
seq(seq(coeff(f(n), x, k), k=0..n), n=0..10); # Robert Israel, Oct 15 2017

Mathematica

(* The function RiordanArray is defined in A256893. *)
RiordanArray[Exp[-ArcTan[Sqrt[3]*#/(# + 2)]/Sqrt[3]]/Sqrt[1 + # + #^2]&, 2*ArcTan[Sqrt[3]*#/(# + 2)]/Sqrt[3]&, 10, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

Crossrefs

Sequence in context: A189610 A172427 A143362 * A210866 A245474 A338369

Adjacent sequences: A182820 A182821 A182822 * A182824 A182825 A182826

Keyword

sign,tabl,look

Author

Paul Barry, Dec 05 2010

Status

approved