A059343 Triangle of nonzero coefficients of Hermite polynomials H_n(x) in increasing powers of x.

1, 2, -2, 4, -12, 8, 12, -48, 16, 120, -160, 32, -120, 720, -480, 64, -1680, 3360, -1344, 128, 1680, -13440, 13440, -3584, 256, 30240, -80640, 48384, -9216, 512, -30240, 302400, -403200, 161280, -23040, 1024, -665280, 2217600, -1774080, 506880, -56320, 2048, 665280, -7983360, 13305600

Offset: 0

Author

N. J. A. Sloane, Jan 27 2001

Keywords

• sign
• easy
• nice
• tabf

Examples

			1; 2*x; -2+4*x^2; -12*x+8*x^3; ...
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 801.
• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 50.

Links

• T. D. Noe, Rows n=0..100 of triangle, flattened
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• P. Diaconis and A. Gamburd, Random matrices, magic squares and matching polynomials
• Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
• Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020.
• Eric Weisstein's World of Mathematics, Hermite Polynomial

Crossrefs

Cf. A059344.

If initial zeros are included, same as A060821.

Programs

Maple

with(orthopoly): h:=proc(n) if n mod 2=0 then expand(x^2*H(n,x)) else expand(x*H(n,x)) fi end: seq(seq(coeff(h(n),x^(2*k)),k=1..1+floor(n/2)),n=0..14); # this gives the signed sequence

Mathematica

Flatten[ Table[ Coefficient[ HermiteH[n, x], x, k], {n, 0, 12}, {k, Mod[n, 2], n, 2}]] (* Jean-François Alcover, Jan 23 2012 *)

Python

from sympy import hermite, Poly, Symbol
x = Symbol('x')
def a(n):
    return Poly(hermite(n, x), x).coeffs()[::-1]
for n in range(21): print(a(n)) # Indranil Ghosh, May 26 2017

Extensions

Edited by Emeric Deutsch, Jun 05 2004