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