A181089 Triangle T(n, k) = A060821(n,k) + A060821(n,n-k), read by rows.

2, 2, 2, 2, 0, 2, 8, -12, -12, 8, 28, 0, -96, 0, 28, 32, 120, -160, -160, 120, 32, -56, 0, 240, 0, 240, 0, -56, 128, -1680, -1344, 3360, 3360, -1344, -1680, 128, 1936, 0, -17024, 0, 26880, 0, -17024, 0, 1936, 512, 30240, -9216, -80640, 48384, 48384, -80640, -9216, 30240, 512

Offset: 0

Author

Roger L. Bagula, Oct 02 2010

Keywords

• sign
• tabl

Examples

			Triangle begins as:
     2;
     2,     2;
     2,     0,      2;
     8,   -12,    -12,      8;
    28,     0,    -96,      0,      28;
    32,   120,   -160,   -160,     120,    32;
   -56,     0,    240,      0,     240,     0,     -56;
   128, -1680,  -1344,   3360,    3360, -1344,   -1680,   128;
  1936,     0, -17024,      0,   26880,     0,  -17024,     0,   1936;
   512, 30240,  -9216, -80640,   48384, 48384,  -80640, -9216,  30240, 512;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A060821, A062267.

Programs

Mathematica

(* First program *)
p[x_, n_] = HermiteH[n, x] + ExpandAll[x^n*HermiteH[n, 1/x]];
Flatten[Table[CoefficientList[p[x, n], x], {n, 0, 15}]] (* edited by G. C. Greubel, Apr 04 2021 *)
(* Second program *)
A060821[n_, k_]:= If[EvenQ[n-k], (-1)^(Floor[(n-k)/2])*2^k*n!/(k!*(Floor[(n - k)/2]!)), 0];
T[n_, k_]:= A060821[n, k] +A060821[n, n-k];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 04 2021 *)

Sage

def A060821(n,k): return (-1)^((n-k)//2)*2^k*factorial(n)/(factorial(k)*factorial( (n-k)//2)) if (n-k)%2==0 else 0
def T(n,k): return A060821(n, k) + A060821(n, n-k)
flatten([[T(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Apr 04 2021

Formula

T(n, k) = coefficients [x^k] of the polynomial HermiteH(n,x) + x^n*HermiteH(n,1/x).

T(n, k) = A060821(n,k) + A060821(n,n-k).

Sum_{k=0..n} T(n, k) = 2*A062267(n).