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A060821 Triangle T(n,k) read by rows giving coefficients of Hermite polynomial of order n (n >= 0, 0 <= k <= n). 25
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

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, Dae San Kim, A note on Hermite polynomials, arXiv:1602.04096 [math.NT], 2016.

Wikipedia, Hermite polynomials

Index entries for sequences related to Hermite polynomials

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;

MATHEMATICA

Flatten[ Table[ CoefficientList[ HermiteH[n, x], x], {n, 0, 10}]] (* Jean-Fran├žois 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

def a(n): return Poly(hermite(n, x), x).all_coeffs()[::-1]

for n in xrange(0, 21): print a(n) # Indranil Ghosh, May 26 2017

CROSSREFS

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

Without initial zeros, same as A059343.

Sequence in context: A138090 A138093 A138094 * A191718 A286777 A286123

Adjacent sequences:  A060818 A060819 A060820 * A060822 A060823 A060824

KEYWORD

sign,tabl,nice

AUTHOR

Vladeta Jovovic, Apr 30 2001

STATUS

approved

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Last modified December 16 09:34 EST 2018. Contains 318160 sequences. (Running on oeis4.)