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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

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OFFSET

0,2

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

EXAMPLE

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

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 *)

PROG

(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

CROSSREFS

Cf.A059344.

If initial zeros are included, same asA060821.