A109767 Triangle T(n,k), 0 <= k <= n, defined by T(n,k) = 2^k*A001497(n,k).

1, 2, 2, 12, 12, 4, 120, 120, 48, 8, 1680, 1680, 720, 160, 16, 30240, 30240, 13440, 3360, 480, 32, 665280, 665280, 302400, 80640, 13440, 1344, 64, 17297280, 17297280, 7983360, 2217600, 403200, 48384, 3584, 128, 518918400, 518918400

Offset

0,2

Comments

Also square array of unsigned coefficients of Hermite polynomials.

T[n,k]is A128099(2n,k)*A001813(n-k). - Richard Turk, Sep 26 2017

Links

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

Formula

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

Example

Rows begin:
     1
     2,    2,
    12,   12,   4,
   120,  120,  48,   8,
  1680, 1680, 720, 160, 16,
Unsigned coefficients of Hermite polynomials:
     1,     2,      4,       8, ...
     2,    12,     48,     160, ...
    12,   120,    720,    3360, ...
   120,  1680,  13440,   80640, ...
  1680, 30240, 302400, 2217600, ...

Maple

seq(seq((2*n-k)!*2^k/(k!*(n-k)!), k=0..n), n=0..10); # Robert Israel, Sep 26 2017

Mathematica

y[n_, x_] := Sqrt[2/(Pi*x)]*E^(1/x)*BesselK[-n-1/2, 1/x]; t[n_, k_] := 2^n*Coefficient[y[n, x], x, k]; Table[t[n, k], {n, 0, 8}, {k, n, 0, -1}] // Flatten (* or *) t[n_, k_] := (2*n - k)!*2^k/(k!*(n-k)!); Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 01 2013 *)
Table[((2n-k)!*2^k)/(k!(n-k)!), {n, 0, 10}, {k, 0, n}]//Flatten (* Harvey P. Dale, Nov 23 2017 *)

Prog

(Magma) /* As triangle */ [[Factorial(2*n-k)*2^k/(Factorial(k)*Factorial(n-k)): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Dec 14 2015

Crossrefs

Cf. A001497.

Sequence in context: A190295 A228154 A275279 * A339297 A196061 A342582

Adjacent sequences: A109764 A109765 A109766 * A109768 A109769 A109770

Keyword

nonn,tabl,nice

Author

Philippe Deléham, Aug 12 2005

Status

approved