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Triangle of nonzero coefficients of Hermite polynomials H_n(x) in increasing powers of x.
12

%I #28 Mar 19 2021 06:57:36

%S 1,2,-2,4,-12,8,12,-48,16,120,-160,32,-120,720,-480,64,-1680,3360,

%T -1344,128,1680,-13440,13440,-3584,256,30240,-80640,48384,-9216,512,

%U -30240,302400,-403200,161280,-23040,1024,-665280,2217600,-1774080,506880,-56320,2048,665280,-7983360,13305600

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

%D 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.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 50.

%H T. D. Noe, <a href="/A059343/b059343.txt">Rows n=0..100 of triangle, flattened</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H P. Diaconis and A. Gamburd, <a href="http://www.combinatorics.org/Volume_11/Abstracts/v11i2r2.html">Random matrices, magic squares and matching polynomials</a>

%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Janjic/janjic22.html">Some classes of numbers and derivatives</a>, JIS 12 (2009) 09.8.3.

%H Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, <a href="https://arxiv.org/abs/2012.04625">Finding structure in sequences of real numbers via graph theory: a problem list</a>, arXiv:2012.04625, Dec 08, 2020.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HermitePolynomial.html">Hermite Polynomial</a>

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

%p 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

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

%o (Python)

%o from sympy import hermite, Poly, Symbol

%o x = Symbol('x')

%o def a(n):

%o return Poly(hermite(n, x), x).coeffs()[::-1]

%o for n in range(21): print(a(n)) # _Indranil Ghosh_, May 26 2017

%Y Cf. A059344.

%Y If initial zeros are included, same as A060821.

%K sign,easy,nice,tabf

%O 0,2

%A _N. J. A. Sloane_, Jan 27 2001

%E Edited by _Emeric Deutsch_, Jun 05 2004