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A109767 Triangle T(n,k), 0 <= k <= n, defined by T(n,k) = 2^k*A001497(n,k). 1

%I

%S 1,2,2,12,12,4,120,120,48,8,1680,1680,720,160,16,30240,30240,13440,

%T 3360,480,32,665280,665280,302400,80640,13440,1344,64,17297280,

%U 17297280,7983360,2217600,403200,48384,3584,128,518918400,518918400

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

%C Also square array of unsigned coefficients of Hermite polynomials.

%C T[n,k]is A128099(2n,k)*A001813(n-k). - _Richard Turk_, Sep 26 2017

%H Robert Israel, <a href="/A109767/b109767.txt">Table of n, a(n) for n = 0..10010</a> (rows 0 to 140, flattened)

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

%e Rows begin:

%e 1

%e 2, 2,

%e 12, 12, 4,

%e 120, 120, 48, 8,

%e 1680, 1680, 720, 160, 16,

%e Unsigned coefficients of Hermite polynomials:

%e 1, 2, 4, 8, ...

%e 2, 12, 48, 160, ...

%e 12, 120, 720, 3360, ...

%e 120, 1680, 13440, 80640, ...

%e 1680, 30240, 302400, 2217600, ...

%p seq(seq((2*n-k)!*2^k/(k!*(n-k)!),k=0..n),n=0..10); # _Robert Israel_, Sep 26 2017

%t 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 *)

%t Table[((2n-k)!*2^k)/(k!(n-k)!),{n,0,10},{k,0,n}]//Flatten (* _Harvey P. Dale_, Nov 23 2017 *)

%o (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

%Y Cf. A001497.

%K nonn,tabl,nice

%O 0,2

%A _Philippe Deléham_, Aug 12 2005

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Last modified December 10 04:15 EST 2019. Contains 329885 sequences. (Running on oeis4.)