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%I #76 Jan 08 2023 11:29:59
%S 1,0,2,-2,0,4,0,-12,0,8,12,0,-48,0,16,0,120,0,-160,0,32,-120,0,720,0,
%T -480,0,64,0,-1680,0,3360,0,-1344,0,128,1680,0,-13440,0,13440,0,-3584,
%U 0,256,0,30240,0,-80640,0,48384,0,-9216,0,512,-30240,0,302400,0,-403200,0,161280,0,-23040,0,1024
%N Triangle read by rows. T(n, k) are the coefficients of the Hermite polynomial of order n, for 0 <= k <= n.
%C Exponential Riordan array [exp(-x^2), 2x]. - _Paul Barry_, Jan 22 2009
%H T. D. Noe, <a href="/A060821/b060821.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, p. 801.
%H Taekyun Kim and Dae San Kim, <a href="http://arxiv.org/abs/1602.04096">A note on Hermite polynomials</a>, arXiv:1602.04096 [math.NT], 2016.
%H Alexander Minakov, <a href="https://arxiv.org/abs/1911.03942">Question about integral of product of four Hermite polynomials integrated with squared weight</a>, arXiv:1911.03942 [math.CO], 2019.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hermite_polynomials">Hermite polynomials</a>
%H <a href="/index/He#Hermite">Index entries for sequences related to Hermite polynomials</a>
%F T(n, k) = ((-1)^((n-k)/2))*(2^k)*n!/(k!*((n-k)/2)!) if n-k is even and >= 0, else 0.
%F E.g.f.: exp(-y^2 + 2*y*x).
%F From _Paul Barry_, Aug 28 2005: (Start)
%F T(n, k) = n!/(k!*2^((n-k)/2)((n-k)/2)!)2^((n+k)/2)cos(Pi*(n-k)/2)(1 + (-1)^(n+k))/2;
%F T(n, k) = A001498((n+k)/2, (n-k)/2)*cos(Pi*(n-k)/2)2^((n+k)/2)(1 + (-1)^(n+k))/2.
%F (End)
%F Row sums: A062267. - _Derek Orr_, Mar 12 2015
%F a(n*(n+3)/2) = a(A000096(n)) = 2^n. - _Derek Orr_, Mar 12 2015
%F Recurrence for fixed n: T(n, k) = -(k+2)*(k+1)/(2*(n-k)) * T(n, k+2), starting with T(n, n) = 2^n. - _Ralf Stephan_, Mar 26 2016
%F The m-th row consecutive nonzero entries in increasing order are (-1)^(c/2)*(c+b)!/(c/2)!b!*2^b with c = m, m-2, ..., 0 and b = m-c if m is even and with c = m-1, m-3, ..., 0 with b = m-c if m is odd. For the 10th row starting at a(55) the 6 consecutive nonzero entries in order are -30240,302400,-403200,161280,-23040,1024 given by c = 10,8,6,4,2,0 and b = 0,2,4,6,8,10. - _Richard Turk_, Aug 20 2017
%e [1], [0, 2], [ -2, 0, 4], [0, -12, 0, 8], [12, 0, -48, 0, 16], [0, 120, 0, -160, 0, 32], ... .
%e Thus H_0(x) = 1, H_1(x) = 2*x, H_2(x) = -2 + 4*x^2, H_3(x) = -12*x + 8*x^3, H_4(x) = 12 - 48*x^2 + 16*x^4, ...
%e Triangle starts:
%e 1;
%e 0, 2;
%e -2, 0, 4;
%e 0, -12, 0, 8;
%e 12, 0, -48, 0, 16;
%e 0, 120, 0, -160, 0, 32;
%e -120, 0, 720, 0, -480, 0, 64;
%e 0, -1680, 0, 3360, 0, -1344, 0, 128;
%e 1680, 0, -13440, 0, 13440, 0, -3584, 0, 256;
%e 0, 30240, 0, -80640, 0, 48384, 0, -9216, 0, 512;
%e -30240, 0, 302400, 0, -403200, 0, 161280, 0, -23040, 0, 1024;
%p with(orthopoly):for n from 0 to 10 do H(n,x):od;
%p T := proc(n,m) if n-m >= 0 and n-m mod 2 = 0 then ((-1)^((n-m)/2))*(2^m)*n!/(m!*((n-m)/2)!) else 0 fi; end;
%p # Alternative:
%p T := proc(n,k) option remember; if k > n then 0 elif n = k then 2^n else
%p (T(n, k+2)*(k+2)*(k+1))/(2*(k-n)) fi end:
%p seq(print(seq(T(n, k), k = 0..n)), n = 0..10); # _Peter Luschny_, Jan 08 2023
%t Flatten[ Table[ CoefficientList[ HermiteH[n, x], x], {n, 0, 10}]] (* _Jean-François Alcover_, Jan 18 2012 *)
%o (PARI) for(n=0,9,v=Vec(polhermite(n));forstep(i=n+1,1,-1,print1(v[i]", "))) \\ _Charles R Greathouse IV_, Jun 20 2012
%o (Python)
%o from sympy import hermite, Poly, symbols
%o x = symbols('x')
%o def a(n): return Poly(hermite(n, x), x).all_coeffs()[::-1]
%o for n in range(21): print(a(n)) # _Indranil Ghosh_, May 26 2017
%o (Python)
%o def Trow(n: int) -> list[int]:
%o row: list[int] = [0] * (n + 1); row[n] = 2**n
%o for k in range(n - 2, -1, -2):
%o row[k] = -(row[k + 2] * (k + 2) * (k + 1)) // (2 * (n - k))
%o return row # _Peter Luschny_, Jan 08 2023
%Y Cf. A001814, A001816, A000321, A062267 (row sums).
%Y Without initial zeros, same as A059343.
%K sign,tabl,nice
%O 0,3
%A _Vladeta Jovovic_, Apr 30 2001
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