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

%I A109767 #37 Feb 16 2022 23:26:44
%S A109767 1,2,2,12,12,4,120,120,48,8,1680,1680,720,160,16,30240,30240,13440,
%T A109767 3360,480,32,665280,665280,302400,80640,13440,1344,64,17297280,
%U A109767 17297280,7983360,2217600,403200,48384,3584,128,518918400,518918400
%N A109767 Triangle T(n,k), 0 <= k <= n, defined by T(n,k) = 2^k*A001497(n,k).
%C A109767 Also square array of unsigned coefficients of Hermite polynomials.
%C A109767 T[n,k]is A128099(2n,k)*A001813(n-k). - _Richard Turk_, Sep 26 2017
%H A109767 Robert Israel, <a href="/A109767/b109767.txt">Table of n, a(n) for n = 0..10010</a> (rows 0 to 140, flattened)
%F A109767 T(n,k) = (2n-k)!*2^k/(k!*(n-k)!).
%e A109767 Rows begin:
%e A109767      1
%e A109767      2,    2,
%e A109767     12,   12,   4,
%e A109767    120,  120,  48,   8,
%e A109767   1680, 1680, 720, 160, 16,
%e A109767 Unsigned coefficients of Hermite polynomials:
%e A109767      1,     2,      4,       8, ...
%e A109767      2,    12,     48,     160, ...
%e A109767     12,   120,    720,    3360, ...
%e A109767    120,  1680,  13440,   80640, ...
%e A109767   1680, 30240, 302400, 2217600, ...
%p A109767 seq(seq((2*n-k)!*2^k/(k!*(n-k)!),k=0..n),n=0..10); # _Robert Israel_, Sep 26 2017
%t A109767 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 A109767 Table[((2n-k)!*2^k)/(k!(n-k)!),{n,0,10},{k,0,n}]//Flatten (* _Harvey P. Dale_, Nov 23 2017 *)
%o A109767 (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 A109767 Cf. A001497.
%K A109767 nonn,tabl,nice
%O A109767 0,2
%A A109767 _Philippe Deléham_, Aug 12 2005