A176231 Coefficient array of orthogonal polynomials whose moment sequence is the double factorial numbers A001147.

1, -1, 1, 3, -6, 1, -15, 45, -15, 1, 105, -420, 210, -28, 1, -945, 4725, -3150, 630, -45, 1, 10395, -62370, 51975, -13860, 1485, -66, 1, -135135, 945945, -945945, 315315, -45045, 3003, -91, 1, 2027025, -16216200, 18918900, -7567560, 1351350, -120120, 5460, -120, 1

Offset: 0

Author

Paul Barry, Apr 12 2010

Keywords

• easy
• sign
• tabl

Comments

Exponential Riordan array [1/sqrt(1+2x),x/(1+2x)]. Inverse of A176230.

Diagonal sums are an alternating sign version of A025164.

Examples

			Triangle begins
  1,
  -1, 1,
  3, -6, 1,
  -15, 45, -15, 1,
  105, -420, 210, -28, 1,
  -945, 4725, -3150, 630, -45, 1,
  10395, -62370, 51975, -13860, 1485, -66, 1,
  -135135, 945945, -945945, 315315, -45045, 3003, -91, 1,
  2027025, -16216200, 18918900, -7567560, 1351350, -120120, 5460, -120, 1
Production matrix is
  -1, 1,
  2, -5, 1,
  0, 12, -9, 1,
  0, 0, 30, -13, 1,
  0, 0, 0, 56, -17, 1,
  0, 0, 0, 0, 90, -21, 1,
  0, 0, 0, 0, 0, 132, -25, 1,
  0, 0, 0, 0, 0, 0, 182, -29, 1,
  0, 0, 0, 0, 0, 0, 0, 240, -33, 1
		

Crossrefs

Cf. A001147, A025164, A176230.

Programs

Maple

T := (n,k) -> (2*n)!*(-1/2)^(n-k)/(2*k)!*(n-k)!:
seq(seq(T(n,k), k=0..n), n=0..8); # Peter Luschny, Jul 20 2019

Mathematica

(* The function RiordanArray is defined in A256893. *)
rows = 9;
R = RiordanArray[1/Sqrt[1 + 2 #]&, #/(1 + 2 #)&, rows, True];
R // Flatten (* Jean-François Alcover, Jul 20 2019 *)
T[ n_, k_] := Coefficient[ HermiteH[2 n, x/Sqrt[2]], x, 2 k]/2^n; (* Michael Somos, Jan 15 2020 *)
T[ n_, k_] := Coefficient[ Nest[# x - D[#, x]&, 1, 2 n], x, 2 k]; (* Michael Somos, Jan 15 2020 *)

PARI

{T(n, k) = my(t=1); for(i=1, 2*n, t = x*t - t'); polcoeff(t, 2*k)}; /* Michael Somos, Jan 15 2020 */

Formula

Number triangle T(n,k) = (-1)^(n-k)*(2n)!/((2k)!(n-k)!2^(n-k)).

He_(2*n)(x) = Sum_{k=0..n} T(n, k)*x^(2*k) where He is Hermite's polynomial. - Michael Somos, Jan 15 2020