A060821 Triangle read by rows. T(n, k) are the coefficients of the Hermite polynomial of order n, for 0 <= k <= n.

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, -480, 0, 64, 0, -1680, 0, 3360, 0, -1344, 0, 128, 1680, 0, -13440, 0, 13440, 0, -3584, 0, 256, 0, 30240, 0, -80640, 0, 48384, 0, -9216, 0, 512, -30240, 0, 302400, 0, -403200, 0, 161280, 0, -23040, 0, 1024

Offset: 0

Author

Vladeta Jovovic, Apr 30 2001

Keywords

• sign
• tabl
• nice

Comments

Exponential Riordan array [exp(-x^2), 2x]. - Paul Barry, Jan 22 2009

Examples

			[1], [0, 2], [ -2, 0, 4], [0, -12, 0, 8], [12, 0, -48, 0, 16], [0, 120, 0, -160, 0, 32], ... .
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, ...
Triangle starts:
     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,    -480,     0,     64;
     0, -1680,      0,   3360,       0, -1344,      0,   128;
  1680,     0, -13440,      0,   13440,     0,  -3584,     0,    256;
     0, 30240,      0, -80640,       0, 48384,      0, -9216,      0, 512;
-30240,     0, 302400,      0, -403200,     0, 161280,     0, -23040,   0, 1024;
		

References

• Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 24, equations 24:4:1 - 24:4:8 at page 219.

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, p. 801.
• Taekyun Kim and Dae San Kim, A note on Hermite polynomials, arXiv:1602.04096 [math.NT], 2016.
• Alexander Minakov, Question about integral of product of four Hermite polynomials integrated with squared weight, arXiv:1911.03942 [math.CO], 2019.
• Wikipedia, Hermite polynomials.
• Index entries for sequences related to Hermite polynomials.

Crossrefs

Cf. A001814, A001816, A000321, A062267 (row sums).

Without initial zeros, same as A059343.

Programs

Maple

with(orthopoly):for n from 0 to 10 do H(n,x):od;
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;
# Alternative:
T := proc(n,k) option remember; if k > n then 0 elif n = k then 2^n else
(T(n, k+2)*(k+2)*(k+1))/(2*(k-n)) fi end:
seq(print(seq(T(n, k), k = 0..n)), n = 0..10); # Peter Luschny, Jan 08 2023

Mathematica

Flatten[ Table[ CoefficientList[ HermiteH[n, x], x], {n, 0, 10}]] (* Jean-François Alcover, Jan 18 2012 *)

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

Python

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

Python

def Trow(n: int) -> list[int]:
    row: list[int] = [0] * (n + 1); row[n] = 2**n
    for k in range(n - 2, -1, -2):
        row[k] = -(row[k + 2] * (k + 2) * (k + 1)) // (2 * (n - k))
    return row  # Peter Luschny, Jan 08 2023

Formula

T(n, k) = ((-1)^((n-k)/2))*(2^k)*n!/(k!*((n-k)/2)!) if n-k is even and >= 0, else 0.

E.g.f.: exp(-y^2 + 2*y*x).

From Paul Barry, Aug 28 2005: (Start)

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;

T(n, k) = A001498((n+k)/2, (n-k)/2)*cos(Pi*(n-k)/2)2^((n+k)/2)(1 + (-1)^(n+k))/2.

(End)

Row sums: A062267. - Derek Orr, Mar 12 2015

a(n*(n+3)/2) = a(A000096(n)) = 2^n. - Derek Orr, Mar 12 2015

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

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