A143507 Triangle of coefficients of x^n*H_n(x + 1/x), where H_n(x) is the Hermite polynomial of order n.
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, -40, 0, -40, 0, 0, 0, 32, 64, 0, -96, 0, -240, 0, -280, 0, -240, 0, -96, 0, 64, 128, 0, -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
Offset: 0
Examples
Triangle begins:
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, -40, 0, -40, 0, 0, 0, 32;
64, 0, -96, 0, -240, 0, -280, 0, -240, 0, -96, 0, 64;
128, 0, -448, 0, -672, 0, -560, 0, -560, 0, -672, 0, -448, 0, 128;
... reformatted. - _Franck Maminirina Ramaharo_, Oct 25 2018
Links
- Wikipedia, Hermite polynomials
Programs
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Mathematica
Table[CoefficientList[FullSimplify[x^n*HermiteH[n, x + 1/x]], x], {n, 0, 10}]//Flatten -
PARI
row(n) = Vec(x^n*subst(polhermite(n,x),x,x+1/x)); for (n=0, 10, print(row(n))); \\ Michel Marcus, Oct 27 2018
Formula
E.g.f.: exp(2*(1 + x^2)*y - x^2*y^2). - Franck Maminirina Ramaharo, Oct 25 2018
Extensions
Edited, new name and offset corrected by Franck Maminirina Ramaharo, Oct 25 2018
Comments