A182824 Inverse of coefficient array for orthogonal polynomials p(n,x)=(x-(2n-1))*p(n-1,x)-(2n-2)^2*p(n-2,x).

1, 1, 1, 5, 4, 1, 21, 33, 9, 1, 153, 264, 114, 16, 1, 1209, 2769, 1410, 290, 25, 1, 12285, 32076, 20259, 5040, 615, 36, 1, 140589, 432657, 314811, 94899, 14175, 1155, 49, 1, 1871217, 6475536, 5423076, 1886304, 337974, 33936, 1988, 64, 1, 27773361, 108067041, 101497860, 40257540, 8321670, 997542, 72324, 3204, 81, 1, 460041525, 1975940244, 2064827781, 915887520, 214906770, 29709288, 2565738, 141120, 4905, 100, 1

Offset: 0

Author

Paul Barry, Dec 05 2010

Keywords

• nonn
• easy
• tabl

Comments

Inverse is the coefficient array for the orthogonal polynomials p(0,x)=1,p(1,x)=x-1,p(n,x)=(x-(2n-1))*p(n-1,x)-(2n-2)^2*p(n-2,x).

Inverse is A182826. First column is A182825.

Examples

			Triangle begins:
  1,
  1, 1,
  5, 4, 1,
  21, 33, 9, 1,
  153, 264, 114, 16, 1,
  1209, 2769, 1410, 290, 25, 1,
  12285, 32076, 20259, 5040, 615, 36, 1,
  140589, 432657, 314811, 94899, 14175, 1155, 49, 1,
  1871217, 6475536, 5423076, 1886304, 337974, 33936, 1988, 64, 1
Production matrix begins:
  1, 1,
  4, 3, 1,
  0, 16, 5, 1,
  0, 0, 36, 7, 1,
  0, 0, 0, 64, 9, 1,
  0, 0, 0, 0, 100, 11, 1,
  0, 0, 0, 0, 0, 144, 13, 1,
  0, 0, 0, 0, 0, 0, 196, 15, 1,
  0, 0, 0, 0, 0, 0, 0, 256, 17, 1
  0, 0, 0, 0, 0, 0, 0, 0, 324, 19, 1
		

Programs

Mathematica

(* The function RiordanArray is defined in A256893. *)
RiordanArray[1/(Cos[Sqrt[3]*#] - Sin[Sqrt[3]*#]/Sqrt[3])&, Sin[Sqrt[3]*#]/ (Sqrt[3]*Cos[Sqrt[3]*#] - Sin[Sqrt[3]*#])&, 11, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

Formula

Exponential Riordan array [1/(cos(sqrt(3)*x)-sin(sqrt(3)*x)/sqrt(3)), sin(sqrt(3)*x)/(sqrt(3)*cos(sqrt(3)*x)-sin(sqrt(3)*x))].