cp's OEIS Frontend

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

%I A059343 #29 Feb 16 2025 08:32:43
%S A059343 1,2,-2,4,-12,8,12,-48,16,120,-160,32,-120,720,-480,64,-1680,3360,
%T A059343 -1344,128,1680,-13440,13440,-3584,256,30240,-80640,48384,-9216,512,
%U A059343 -30240,302400,-403200,161280,-23040,1024,-665280,2217600,-1774080,506880,-56320,2048,665280,-7983360,13305600
%N A059343 Triangle of nonzero coefficients of Hermite polynomials H_n(x) in increasing powers of x.
%D A059343 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 A059343 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 50.
%H A059343 T. D. Noe, <a href="/A059343/b059343.txt">Rows n=0..100 of triangle, flattened</a>
%H A059343 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 A059343 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 A059343 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 A059343 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 A059343 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HermitePolynomial.html">Hermite Polynomial</a>
%e A059343 1; 2*x; -2+4*x^2; -12*x+8*x^3; ...
%p A059343 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 A059343 Flatten[ Table[ Coefficient[ HermiteH[n, x], x, k], {n, 0, 12}, {k, Mod[n, 2], n, 2}]] (* _Jean-François Alcover_, Jan 23 2012 *)
%o A059343 (Python)
%o A059343 from sympy import hermite, Poly, Symbol
%o A059343 x = Symbol('x')
%o A059343 def a(n):
%o A059343     return Poly(hermite(n, x), x).coeffs()[::-1]
%o A059343 for n in range(21): print(a(n)) # _Indranil Ghosh_, May 26 2017
%Y A059343 Cf. A059344.
%Y A059343 If initial zeros are included, same as A060821.
%K A059343 sign,easy,nice,tabf
%O A059343 0,2
%A A059343 _N. J. A. Sloane_, Jan 27 2001
%E A059343 Edited by _Emeric Deutsch_, Jun 05 2004