A067613 Triangular table of coefficients of the Hermite polynomials, divided by 2^floor(n/2).

1, 0, -2, -1, 0, 2, 0, 6, 0, -4, 3, 0, -12, 0, 4, 0, -30, 0, 40, 0, -8, -15, 0, 90, 0, -60, 0, 8, 0, 210, 0, -420, 0, 168, 0, -16, 105, 0, -840, 0, 840, 0, -224, 0, 16, 0, -1890, 0, 5040, 0, -3024, 0, 576, 0, -32, -945, 0, 9450, 0, -12600, 0, 5040, 0, -720, 0, 32, 0, 20790, 0, -69300, 0, 55440, 0, -15840, 0, 1760, 0, -64

Offset: 0

Author

Wouter Meeussen, Feb 01 2002

Keywords

• easy
• sign
• tabl
• look

Comments

Series development of exp(-(c+x)^2) at x=0 gives a Hermite polynomial in c as coefficient for x^k.

Links

• Robert Israel, Table of n, a(n) for n = 0..10010(rows 0 to 140, flattened)

Crossrefs

Cf. A060821.

Programs

Maple

S:=series(exp(-2*c*x-x^2),x,13):
seq(seq(coeff(coeff(S,x,n)*n!/2^floor(n/2),c,j),j=0..n),n=0..12); # Robert Israel, Dec 07 2018

Mathematica

Table[ CoefficientList[ HermiteH[ n, c ], c ](-1)^n/2^Floor[ n/2 ], {n, 0, 12} ] (* or, equivalently *) a1=CoefficientList[ Series[ Exp[ c^2 ]Exp[ -(c+x)^2 ], {x, 0, 12} ], x ]; a2=(CoefficientList[ #, c ]&/@ a1 ) Range[ 0, 12 ]! 2^-Floor[ Range[ 0, 12 ]/2 ]

PARI

row(n) = Vecrev((-1)^n*polhermite(n)/2^floor(n/2)) \\ Michel Marcus, Dec 07 2018

Formula

HermiteH[n, c](-1)^n / 2^Floor[n/2]