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

%I A181089 #8 Apr 05 2021 00:07:16
%S A181089 2,2,2,2,0,2,8,-12,-12,8,28,0,-96,0,28,32,120,-160,-160,120,32,-56,0,
%T A181089 240,0,240,0,-56,128,-1680,-1344,3360,3360,-1344,-1680,128,1936,0,
%U A181089 -17024,0,26880,0,-17024,0,1936,512,30240,-9216,-80640,48384,48384,-80640,-9216,30240,512
%N A181089 Triangle T(n, k) = A060821(n,k) + A060821(n,n-k), read by rows.
%H A181089 G. C. Greubel, <a href="/A181089/b181089.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A181089 T(n, k) = coefficients [x^k] of the polynomial HermiteH(n,x) + x^n*HermiteH(n,1/x).
%F A181089 T(n, k) = A060821(n,k) + A060821(n,n-k).
%F A181089 Sum_{k=0..n} T(n, k) = 2*A062267(n).
%e A181089 Triangle begins as:
%e A181089      2;
%e A181089      2,     2;
%e A181089      2,     0,      2;
%e A181089      8,   -12,    -12,      8;
%e A181089     28,     0,    -96,      0,      28;
%e A181089     32,   120,   -160,   -160,     120,    32;
%e A181089    -56,     0,    240,      0,     240,     0,     -56;
%e A181089    128, -1680,  -1344,   3360,    3360, -1344,   -1680,   128;
%e A181089   1936,     0, -17024,      0,   26880,     0,  -17024,     0,   1936;
%e A181089    512, 30240,  -9216, -80640,   48384, 48384,  -80640, -9216,  30240, 512;
%t A181089 (* First program *)
%t A181089 p[x_, n_] = HermiteH[n, x] + ExpandAll[x^n*HermiteH[n, 1/x]];
%t A181089 Flatten[Table[CoefficientList[p[x, n], x], {n, 0, 15}]] (* edited by _G. C. Greubel_, Apr 04 2021 *)
%t A181089 (* Second program *)
%t A181089 A060821[n_, k_]:= If[EvenQ[n-k], (-1)^(Floor[(n-k)/2])*2^k*n!/(k!*(Floor[(n - k)/2]!)), 0];
%t A181089 T[n_, k_]:= A060821[n, k] +A060821[n, n-k];
%t A181089 Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 04 2021 *)
%o A181089 (Sage)
%o A181089 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
%o A181089 def T(n,k): return A060821(n, k) + A060821(n, n-k)
%o A181089 flatten([[T(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Apr 04 2021
%Y A181089 Cf. A060821, A062267.
%K A181089 sign,tabl
%O A181089 0,1
%A A181089 _Roger L. Bagula_, Oct 02 2010