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A060821 Triangle read by rows. T(n, k) are the coefficients of the Hermite polynomial of order n, for 0 <= k <= n. 30
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Exponential Riordan array [exp(-x^2), 2x]. - Paul Barry, Jan 22 2009
LINKS
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.
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
EXAMPLE
[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;
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 *)
PROG
(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
CROSSREFS
Cf. A001814, A001816, A000321, A062267 (row sums).
Without initial zeros, same as A059343.
Sequence in context: A138090 A138093 A138094 * A191718 A286777 A286123
KEYWORD
sign,tabl,nice
AUTHOR
Vladeta Jovovic, Apr 30 2001
STATUS
approved

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Last modified September 30 02:05 EDT 2023. Contains 365781 sequences. (Running on oeis4.)