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A143507 Triangle of coefficients of x^n*H_n(x + 1/x), where H_n(x) is the Hermite polynomial of order n.

%I A143507 #13 Oct 27 2018 06:20:25
%S A143507 1,2,0,2,4,0,6,0,4,8,0,12,0,12,0,8,16,0,16,0,12,0,16,0,16,32,0,0,0,
%T A143507 -40,0,-40,0,0,0,32,64,0,-96,0,-240,0,-280,0,-240,0,-96,0,64,128,0,
%U A143507 -448,0,-672,0,-560,0,-560,0,-672,0,-448,0,128,256,0,-1536,0,-896,0,896,0,1680,0,896,0,-896,0,-1536,0,256,512,0,-4608,0,512
%N A143507 Triangle of coefficients of x^n*H_n(x + 1/x), where H_n(x) is the Hermite polynomial of order n.
%C A143507 Row sums yield A144141.
%H A143507 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hermite_polynomials">Hermite polynomials</a>
%F A143507 E.g.f.: exp(2*(1 + x^2)*y - x^2*y^2). - _Franck Maminirina Ramaharo_, Oct 25 2018
%e A143507 Triangle begins:
%e A143507      1;
%e A143507      2, 0,    2;
%e A143507      4, 0,    6, 0,    4;
%e A143507      8, 0,   12, 0,   12, 0,    8;
%e A143507     16, 0,   16, 0,   12, 0,   16, 0,   16;
%e A143507     32, 0,    0, 0,  -40, 0,  -40, 0,    0, 0,   32;
%e A143507     64, 0,  -96, 0, -240, 0, -280, 0, -240, 0,  -96, 0,   64;
%e A143507    128, 0, -448, 0, -672, 0, -560, 0, -560, 0, -672, 0, -448, 0, 128;
%e A143507     ... reformatted. - _Franck Maminirina Ramaharo_, Oct 25 2018
%t A143507 Table[CoefficientList[FullSimplify[x^n*HermiteH[n, x + 1/x]], x], {n,
%t A143507   0, 10}]//Flatten
%o A143507 (PARI) row(n) = Vec(x^n*subst(polhermite(n,x),x,x+1/x));
%o A143507 for (n=0, 10, print(row(n))); \\ _Michel Marcus_, Oct 27 2018
%Y A143507 Cf. A060821.
%Y A143507 Cf. A143505, A143506.
%K A143507 sign,tabf
%O A143507 0,2
%A A143507 _Roger L. Bagula_ and _Gary W. Adamson_, Oct 25 2008
%E A143507 Edited, new name and offset corrected by _Franck Maminirina Ramaharo_, Oct 25 2018