A059343 -id:A059343 - cp's OEIS frontend

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

Vladeta Jovovic, Apr 30 2001

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

			[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;

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.

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.

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

Without initial zeros, same as A059343.

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

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

for(n=0,9,v=Vec(polhermite(n));forstep(i=n+1,1,-1,print1(v[i]", "))) \\ Charles R Greathouse IV, Jun 20 2012

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

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

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

A099174 Triangle read by rows: coefficients of modified Hermite polynomials.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 3, 0, 6, 0, 1, 0, 15, 0, 10, 0, 1, 15, 0, 45, 0, 15, 0, 1, 0, 105, 0, 105, 0, 21, 0, 1, 105, 0, 420, 0, 210, 0, 28, 0, 1, 0, 945, 0, 1260, 0, 378, 0, 36, 0, 1, 945, 0, 4725, 0, 3150, 0, 630, 0, 45, 0, 1, 0, 10395, 0, 17325, 0, 6930, 0, 990, 0, 55, 0, 1

Offset: 0

Ralf Stephan, on a suggestion of Karol A. Penson, Oct 13 2004

Absolute values of A066325.
T(n,k) is the number of involutions of {1,2,...,n}, having k fixed points (0 <= k <= n). Example: T(4,2)=6 because we have 1243,1432,1324,4231,3214 and 2134. - Emeric Deutsch, Oct 14 2006
Riordan array [exp(x^2/2),x]. - Paul Barry, Nov 06 2008
Same as triangle of Bessel numbers of second kind, B(n,k) (see Cheon et al., 2013). - N. J. A. Sloane, Sep 03 2013
The modified Hermite polynomial h(n,x) (as in the Formula section) is the numerator of the rational function given by f(n,x) = x + (n-2)/f(n-1,x), where f(x,0) = 1. - Clark Kimberling, Oct 20 2014
Second lower diagonal T(n,n-2) equals positive triangular numbers A000217 \ {0}. - M. F. Hasler, Oct 23 2014
From James East, Aug 17 2015: (Start)
T(n,k) is the number of R-classes (equivalently, L-classes) in the D-class consisting of all rank k elements of the Brauer monoid of degree n.
For n < k with n == k (mod 2), T(n,k) is the rank (minimal size of a generating set) and idempotent rank (minimal size of an idempotent generating set) of the ideal consisting of all rank <= k elements of the Brauer monoid. (End)
This array provides the coefficients of a Laplace-dual sequence H(n,x) of the Dirac delta function, delta(x), and its derivatives, formed by taking the inverse Laplace transform of these modified Hermite polynomials. H(n,x) = h(n,D) delta(x) with h(n,x) as in the examples and the lowering and raising operators L = -x and R = -x + D = -x + d/dx such that L H(n,x) = n * H(n-1,x) and R H(n,x) = H(n+1,x). The e.g.f. is exp[t H(.,x)] = e^(t^2/2) e^(t D) delta(x) = e^(t^2/2) delta(x+t). - Tom Copeland, Oct 02 2016
Antidiagonals of this entry are rows of A001497. - Tom Copeland, Oct 04 2016
This triangle is the reverse of that in Table 2 on p. 7 of the Artioli et al. paper and Table 6.2 on p. 234 of Licciardi's thesis, with associations to the telephone numbers. - Tom Copeland, Jun 18 2018 and Jul 08 2018
See A344678 for connections to a Heisenberg-Weyl algebra of differential operators, matching and independent edge sets of the regular n-simplices with partially labeled vertices, and telephone switchboard scenarios. - Tom Copeland, Jun 02 2021

			h(0,x) = 1
h(1,x) = x
h(2,x) = x^2 + 1
h(3,x) = x^3 + 3*x
h(4,x) = x^4 + 6*x^2 + 3
h(5,x) = x^5 + 10*x^3 + 15*x
h(6,x) = x^6 + 15*x^4 + 45*x^2 + 15
From _Paul Barry_, Nov 06 2008: (Start)
Triangle begins
   1,
   0,  1,
   1,  0,  1,
   0,  3,  0,  1,
   3,  0,  6,  0,  1,
   0, 15,  0, 10,  0,  1,
  15,  0, 45,  0, 15,  0,  1
Production array starts
  0, 1,
  1, 0, 1,
  0, 2, 0, 1,
  0, 0, 3, 0, 1,
  0, 0, 0, 4, 0, 1,
  0, 0, 0, 0, 5, 0, 1 (End)

Alois P. Heinz, Rows n = 0..150, flattened
M. Artioli, G. Dattoli, S. Licciardi, and S. Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017.
Paul Barry, Riordan array, orthogonal polynomials as moments, and Hankel transforms, arXiv:1102.0921 [math.CO], 2011.
G.-S. Cheon, J.-H. Jung and L. W. Shapiro, Generalized Bessel numbers and some combinatorial settings, Discrete Math., 313 (2013), 2127-2138.
T. Copeland, Juggling Zeros in the Matrix (Example II), 2020.
James East and Robert D. Gray, Diagram monoids and Graham-Houghton graphs: idempotents and generating sets of ideals, arXiv:1404.2359 [math.GR], 2014. See Theorem 8.4 and Table 7. - _James East_, Aug 17 2015
A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, A product formula and combinatorial field theory, arXiv:quant-ph/0409152, 2004.
Alexander Kreinin, Combinatorial Properties of Mills' Ratio, arXiv:1405.5852 [math.CO], 2014. See Table 2. - _N. J. A. Sloane_, May 29 2014
S. Licciardi, Umbral Calculus, a Different Mathematical Language, arXiv:1803.03108 [math.CA], 2018.
R. Paris, A uniform asymptotic expansion for the incomplete gamma function, Journal of Computational and Applied Mathematics, 148 (2002), p. 223-239 (See p. 329 and A137286. From _Tom Copeland_, Jan 03 2016).
R. Sazdanovic, A categorification of the polynomial ring, slide presentation, 2011
S. Yang and Z. Qiao, The Bessel numbers and Bessel matrices, Jrn. Math. Rsch. and Exposition, July 2011, Vol. 31, No. 4, pp.627-636. DOI:10.3770/j.issn:1000-341X.2011.04.006.

Row sums (polynomial values at x=1) are A000085.
Polynomial values: A005425 (x=2), A202834 (x=3), A202879(x=4).
Cf. A000217, A001147, A059343, A060821, A066325.
Cf. A000384, A014105, A034839, A049403, A096713, A100861, A104556, A122848, A130757, A176230, A176231.
Cf. A137286.
Cf. A001497.
Cf. A111062, A344678.

T:=proc(n,k) if n-k mod 2 = 0 then n!/2^((n-k)/2)/((n-k)/2)!/k! else 0 fi end: for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form; Emeric Deutsch, Oct 14 2006

nn=10;a=y x+x^2/2!;Range[0,nn]!CoefficientList[Series[Exp[a],{x,0,nn}],{x,y}]//Grid  (* Geoffrey Critzer, May 08 2012 *)
H[0, x_] = 1; H[1, x_] := x; H[n_, x_] := H[n, x] = x*H[n-1, x]-(n-1)* H[n-2, x]; Table[CoefficientList[H[n, x], x], {n, 0, 11}] // Flatten // Abs (* Jean-François Alcover, May 23 2016 *)
T[ n_, k_] := If[ n < 0, 0, Coefficient[HermiteH[n, x I/Sqrt[2]] (Sqrt[1/2]/I)^n, x, k]]; (* Michael Somos, May 10 2019 *)

T(n,k)=if(k<=n && k==Mod(n,2), n!/k!/(k=(n-k)/2)!>>k) \\ M. F. Hasler, Oct 23 2014

import sympy
from sympy import Poly
from sympy.abc import x, y
def H(n, x): return 1 if n==0 else x if n==1 else x*H(n - 1, x) - (n - 1)*H(n - 2, x)
def a(n): return [abs(cf) for cf in Poly(H(n, x), x).all_coeffs()[::-1]]
for n in range(21): print(a(n)) # Indranil Ghosh, May 26 2017

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

def A099174_triangle(dim):
    M = matrix(ZZ,dim,dim)
    for n in (0..dim-1): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1]+(k+1)*M[n-1,k+1]
    return M
A099174_triangle(9)  # Peter Luschny, Oct 06 2012

h(k, x) = (-I/sqrt(2))^k * H(k, I*x/sqrt(2)), H(n, x) the Hermite polynomials (A060821, A059343).
T(n,k) = n!/(2^((n-k)/2)*((n-k)/2)!k!) if n-k >= 0 is even; 0 otherwise. - Emeric Deutsch, Oct 14 2006
G.f.: 1/(1-x*y-x^2/(1-x*y-2*x^2/(1-x*y-3*x^2/(1-x*y-4*x^2/(1-... (continued fraction). - Paul Barry, Apr 10 2009
E.g.f.: exp(y*x + x^2/2). - Geoffrey Critzer, May 08 2012
Recurrence: T(0,0)=1, T(0,k)=0 for k>0 and for n >= 1 T(n,k) = T(n-1,k-1) + (k+1)*T(n-1,k+1). - Peter Luschny, Oct 06 2012
T(n+2,n) = A000217(n+1), n >= 0. - M. F. Hasler, Oct 23 2014
The row polynomials P(n,x) = (a. + x)^n, umbrally evaluated with (a.)^n = a_n = aerated A001147, are an Appell sequence with dP(n,x)/dx = n * P(n-1,x). The umbral compositional inverses (cf. A001147) of these polynomials are given by the same polynomials signed, A066325. - Tom Copeland, Nov 15 2014
From Tom Copeland, Dec 13 2015: (Start)
The odd rows are (2x^2)^n x n! L(n,-1/(2x^2),1/2), and the even, (2x^2)^n n! L(n,-1/(2x^2),-1/2) in sequence with n= 0,1,2,... and L(n,x,a) = Sum_{k=0..n} binomial(n+a,k+a) (-x)^k/k!, the associated Laguerre polynomial of order a. The odd rows are related to A130757, and the even to A176230 and A176231. Other versions of this entry are A122848, A049403, A096713 and A104556, and reversed A100861, A144299, A111924. With each non-vanishing diagonal divided by its initial element A001147(n), this array becomes reversed, aerated A034839.
Create four shift and stretch matrices S1,S2,S3, and S4 with all elements zero except S1(2n,n) = 1 for n >= 1, S2(n,2n) = 1 for n >= 0, S3(2n+1,n) = 1 for n >= 1, and S4(n,2n+1) = 1 for n >= 0. Then this entry's lower triangular matrix is T = Id + S1 * (A176230-Id) * S2 + S3 * (unsigned A130757-Id) * S4 with Id the identity matrix. The sandwiched matrices have infinitesimal generators with the nonvanishing subdiagonals A000384(n>0) and A014105(n>0).
As an Appell sequence, the lowering and raising operators are L = D and R = x + dlog(exp(D^2/2))/dD = x + D, where D = d/dx, L h(n,x) = n h(n-1,x), and R h(n,x) = h(n+1,x), so R^n 1 = h(n,x). The fundamental moment sequence has the e.g.f. e^(t^2/2) with coefficients a(n) = aerated A001147, i.e., h(n,x) = (a. + x)^n, as noted above. The raising operator R as a matrix acting on o.g.f.s (formal power series) is the transpose of the production matrix P below, i.e., (1,x,x^2,...)(P^T)^n (1,0,0,...)^T = h(n,x).
For characterization as a Riordan array and associations to combinatorial structures, see the Barry link and the Yang and Qiao reference. For relations to projective modules, see the Sazdanovic link.
(End)
From the Appell formalism, e^(D^2/2) x^n = h_n(x), the n-th row polynomial listed below, and e^(-D^2/2) x^n = u_n(x), the n-th row polynomial of A066325. Then R = e^(D^2/2) * x * e^(-D^2/2) is another representation of the raising operator, implied by the umbral compositional inverse relation h_n(u.(x)) = x^n. - Tom Copeland, Oct 02 2016
h_n(x) = p_n(x-1), where p_n(x) are the polynomials of A111062, related to the telephone numbers A000085. - Tom Copeland, Jun 26 2018
From Tom Copeland, Jun 06 2021: (Start)
In the power basis x^n, the matrix infinitesimal generator M = A132440^2/2, when acting on a row vector for an o.g.f., is the matrix representation for the differential operator D^2/2.
e^{M} gives the coefficients of the Hermite polynomials of this entry.
The only nonvanishing subdiagonal of M, the second subdiagonal (1,3,6,10,...), gives, aside from the initial 0, the triangular numbers A000217, the number of edges of the n-dimensional simplices with (n+1) vertices. The perfect matchings of these simplices are the aerated odd double factorials A001147 noted above, the moments for the Hermite polynomials.
The polynomials are also generated from A036040 with x[1] = x, x[2] = 1, and the other indeterminates equal to zero. (End)

A059344 Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 1, 12, 12, 1, 20, 60, 1, 30, 180, 120, 1, 42, 420, 840, 1, 56, 840, 3360, 1680, 1, 72, 1512, 10080, 15120, 1, 90, 2520, 25200, 75600, 30240, 1, 110, 3960, 55440, 277200, 332640, 1, 132, 5940, 110880, 831600, 1995840, 665280, 1, 156

Offset: 0

N. J. A. Sloane, Jan 27 2001

			Triangle begins
  1;
  1;
  1,     2;
  1,     6;
  1,    12,    12;
  1,    20,    60;
  1,    30,   180,   120;
  1,    42,   420,   840;
  1,    56,   840,  3360,  1680;
  1,    72,  1512, 10080, 15120;
x^2 = 1/2^2*(Hermite(2,x)+2*Hermite(0,x)); x^3 = 1/2^3*(Hermite(3,x)+6*Hermite(1,x)); x^4 = 1/2^4*(Hermite(4,x)+12*Hermite(2,x)+12*Hermite(0,x)); x^5 = 1/2^5*(Hermite(5,x)+20*Hermite(3,x)+60*Hermite(1,x)); x^6 = 1/2^6*(Hermite(6,x)+30*Hermite(4,x)+180*Hermite(2,x)+120*Hermite(0,x)). - _Vladeta Jovovic_, Feb 21 2003
1 = H(0); 2x = H(1); 4x^2 = H(2)+2H(0); 8x^3 = H(3)+6H(1); etc. where H(k)=Hermite(k,x).

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.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 50.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.

Cf. A059343, A060821.

Cf. A119275 (signed row reverse).

Flatten[Table[n!/(k! * (n-2k)!), {n, 0, 13}, {k, 0, Floor[n/2]}]]
(* Second program: *)
row[n_] := Table[h[k], {k, n, Mod[n, 2], -2}] /. SolveAlways[2^n*x^n == Sum[h[k]*HermiteH[k, x], {k, Mod[n, 2], n, 2}], x] // First; Table[ row[n], {n, 0, 13}] // Flatten (* Jean-François Alcover, Jan 05 2016 *)

for(n=0,25, for(k=0,floor(n/2), print1(n!/(k!*(n-2*k)!), ", "))) \\ G. C. Greubel, Jan 07 2017

E.g.f.: exp(x^2+y*x). - Vladeta Jovovic, Feb 21 2003

a(n, k) = n!/(k! (n-2k)!). - Dean Hickerson, Feb 24 2003

More terms from Vladeta Jovovic, Feb 21 2003

Edited by Emeric Deutsch, Jun 05 2004

A277281 Maximal coefficient (ignoring signs) in Hermite polynomial of order n.

Original entry on oeis.org

1, 2, 4, 12, 48, 160, 720, 3360, 13440, 80640, 403200, 2217600, 13305600, 69189120, 484323840, 2905943040, 19372953600, 131736084480, 846874828800, 6436248698880, 42908324659200, 337903056691200, 2477955749068800, 18997660742860800, 151981285942886400

Offset: 0

Vladimir Reshetnikov, Oct 08 2016

nonn

			For n = 5, H_5(x) = 32*x^5 - 160*x^3 + 120*x. The maximal coefficient (ignoring signs) is 160, so a(5) = 160.

Vaclav Kotesovec, Table of n, a(n) for n = 0..715
Eric Weisstein's World of Mathematics, Hermite Polynomial.
Wikipedia, Hermite polynomials.

Cf. A059343, A277280 (with signs).

Table[Max@Abs@CoefficientList[HermiteH[n, x], x], {n, 0, 25}]

a(n) = vecmax(apply(x->abs(x), Vec(polhermite(n)))); \\ Michel Marcus, Oct 09 2016

from sympy import hermite, Poly
def a(n): return max(map(abs, Poly(hermite(n, x), x).coeffs())) # Indranil Ghosh, May 26 2017

A277280 Maximal coefficient in Hermite polynomial of order n.

Original entry on oeis.org

1, 2, 4, 8, 16, 120, 720, 3360, 13440, 48384, 302400, 2217600, 13305600, 69189120, 322882560, 2421619200, 19372953600, 131736084480, 790416506880, 4290832465920, 40226554368000, 337903056691200, 2477955749068800, 16283709208166400, 113985964457164800

Offset: 0

Vladimir Reshetnikov, Oct 08 2016

nonn

			For n = 5, H_5(x) = 32*x^5 - 160*x^3 + 120*x. The maximal coefficient is 120 (we take signs into account, so -160 < 120), hence a(5) = 120.

Vaclav Kotesovec, Table of n, a(n) for n = 0..710
Vaclav Kotesovec, Plot of a(n+1)/(a(n)*sqrt(n)) for n = 1..10000
Eric Weisstein's World of Mathematics, Hermite Polynomial.
Wikipedia, Hermite polynomials.

Cf. A059343, A277281 (ignoring signs).

Table[Max@CoefficientList[HermiteH[n, x], x], {n, 0, 25}]

a(n) = vecmax(Vec(polhermite(n))); \\ Michel Marcus, Oct 09 2016

from sympy import hermite, Poly
def a(n): return max(Poly(hermite(n, x), x).coeffs()) # Indranil Ghosh, May 26 2017

A163322 The 3rd Hermite Polynomial evaluated at n: H_3(n) = 8n^3 - 12n.

Original entry on oeis.org

0, -4, 40, 180, 464, 940, 1656, 2660, 4000, 5724, 7880, 10516, 13680, 17420, 21784, 26820, 32576, 39100, 46440, 54644, 63760, 73836, 84920, 97060, 110304, 124700, 140296, 157140, 175280, 194764, 215640, 237956, 261760, 287100, 314024, 342580

Offset: 0

Vincenzo Librandi, Jul 25 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Hermite polynomials
Eric Weisstein's World of Mathematics, Hermite Polynomial.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A060821, A059343.

[8*n^3-12*n: n in [0..40]]; // Vincenzo Librandi, Mar 05 2012

A163322 := proc(n) orthopoly[H](3,n) ; end: seq(A163322(n),n=0..80) ; # R. J. Mathar, Jul 26 2009

CoefficientList[Series[-4*x*(1-14*x+x^2)/(x-1)^4,{x,0,40}],x] (* Vincenzo Librandi, Mar 05 2012 *)
LinearRecurrence[{4,-6,4,-1},{0,-4,40,180},40] (* Harvey P. Dale, Aug 14 2014 *)

a(n)=8*n^3-12*n \\ Charles R Greathouse IV, Jan 29 2016

from sympy import hermite
def A163322(n): return hermite(3,n) # Chai Wah Wu, Jan 06 2022

a(n) = 8*n^3 - 12*n.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: -4*x*(1-14*x+x^2)/(x-1)^4.

Edited by R. J. Mathar, Jul 26 2009

A163323 The 4th Hermite Polynomial evaluated at n: H_4(n) = 16n^4 - 48n^2 + 12.

Original entry on oeis.org

12, -20, 76, 876, 3340, 8812, 19020, 36076, 62476, 101100, 155212, 228460, 324876, 448876, 605260, 799212, 1036300, 1322476, 1664076, 2067820, 2540812, 3090540, 3724876, 4452076, 5280780, 6220012, 7279180, 8468076, 9796876, 11276140

Offset: 0

Vincenzo Librandi, Jul 25 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Hermite Polynomial
Index entries for sequences related to Hermite polynomials
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A059343, A060821, A163322.

[16*n^4-48*n^2+12: n in [0..40]]; // Vincenzo Librandi, Mar 05 2012

A163323 := proc(n) orthopoly[H](4,n) ; end: seq(A163323(n),n=0..80) ; # R. J. Mathar, Jul 26 2009

Table[HermiteH[4,n],{n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *)
Table[16 n^4 - 48 n^2 + 12, {n, 0, 30}] (* Vincenzo Librandi, Sep 25 2014 *)
LinearRecurrence[{5,-10,10,-5,1},{12,-20,76,876,3340},40] (* Harvey P. Dale, Jul 03 2019 *)

a(n)=16*n^4-48*n^2+12 \\ Charles R Greathouse IV, Jan 29 2016

from sympy import hermite
def A163323(n): return hermite(4,n) # Chai Wah Wu, Jan 06 2022

a(n) = 16*n^4 - 48*n^2 + 12.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: 4*(-3 +20*x -74*x^2 -44*x^3 +5*x^4)/(x-1)^5.
H_(m+1)(x) = 2*x*H_m(x) - 2*m*H_(m-1)(x), with H_0(x)=1, H_1(x)=2x.

Edited by R. J. Mathar, Jul 26 2009

A277378 Expansion of e.g.f. exp(2*x/(1-x))/sqrt(1-x^2).

Original entry on oeis.org

1, 2, 9, 50, 361, 3042, 29929, 331298, 4100625, 55777922, 828691369, 13316140818, 230256982201, 4257449540450, 83834039024649, 1750225301567618, 38614608429012001, 897325298084953602, 21904718673762721225, 560258287738117292018, 14981472258320814527241

Offset: 0

Vladimir Reshetnikov, Oct 11 2016

nonn

Eric Weisstein's World of Mathematics, Hermite Polynomial.
Wikipedia, Hermite polynomials.

Cf. A000321, A000898, A059343, A062267, A067994, A277280, A277281.

Table[Abs[HermiteH[n, I]]^2/2^n, {n, 0, 20}]
With[{nn=20},CoefficientList[Series[Exp[2x/(1-x)]/Sqrt[1-x^2],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jan 27 2023 *)

E.g.f.: exp(2*x/(1-x))/sqrt(1-x^2).
a(n) = |H_n(i)|^2 / 2^n = H_n(i) * H_n(-i) / 2^n, where H_n(x) is n-th Hermite polynomial, i = sqrt(-1).
D-finite with recurrence: (n+2)*(a(n) + n*a(n-1)) = a(n+1) + n*(n-1)^2*a(n-2).
a(n) ~ n^n / (2 * exp(1 - 2*sqrt(2*n) + n)) * (1 + 2*sqrt(2)/(3*sqrt(n))). - Vaclav Kotesovec, Oct 27 2021

A185296 Triangle of connection constants between the falling factorials (x)(n) and (2*x)(n).

Original entry on oeis.org

1, 0, 2, 0, 2, 4, 0, 0, 12, 8, 0, 0, 12, 48, 16, 0, 0, 0, 120, 160, 32, 0, 0, 0, 120, 720, 480, 64, 0, 0, 0, 0, 1680, 3360, 1344, 128, 0, 0, 0, 0, 1680, 13440, 13440, 3584, 256, 0, 0, 0, 0, 0, 30240, 80640, 48384, 9216, 512

Offset: 0

Peter Bala, Feb 20 2011

The falling factorial polynomials (x)_n := x*(x-1)*...*(x-n+1), n = 0,1,2,..., form a basis for the space of polynomials. Hence the polynomial (2*x)_n may be expressed as a linear combination of x_0, x_1,...,x_n; the coefficients in the expansion form the n-th row of the table. Some examples are given below.
This triangle is connected to two families of orthogonal polynomials, the Hermite polynomials H(n,x) A060821, and the Bessel polynomials y(n,x)A001498. The first few Hermite polynomials are
... 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.
The unsigned coefficients of H(n,x) give the nonzero entries of the n-th row of the triangle.
The Bessel polynomials y(n,x) begin
... y(0,x) = 1
... y(1,x) = 1+x
... y(2,x) = 1+3*x+3*x^2
... y(3,x) = 1+6*x+15*x^2+15*x^3.
The entries in the n-th column of this triangle are the coefficients of the scaled Bessel polynomials 2^n*y(n,x).
Also the Bell transform of g(n) = 2 if n<2 else 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

			Triangle begins
n\k|...0.....1.....2.....3.....4.....5.....6
============================================
0..|...1
1..|...0.....2
2..|...0.....2.....4
3..|...0.....0....12.....8
4..|...0.....0....12....48....16
5..|...0.....0.....0...120...160....32
6..|...0.....0.....0...120...720...480....64
..
Row 3:
(2*x)_3 = (2*x)*(2*x-1)*(2*x-2) = 8*x*(x-1)*(x-2) + 12*x*(x-1).
Row 4:
(2*x)_4 = (2*x)*(2*x-1)*(2*x-2)*(2*x-3) = 16*x*(x-1)*(x-2)*(x-3) +
48*x*(x-1)*(x-2)+ 12*x*(x-1).
Examples of recurrence relation
T(4,4) = 5*T(3,4) + 2*T(3,3) = 5*0 + 2*8 = 16;
T(5,4) = 4*T(4,4) + 2*T(4,3) = 4*16 + 2*48 = 160;
T(6,4) = 3*T(5,4) + 2*T(5,3) = 3*160 + 2*120 = 720;
T(7,4) = 2*T(6,4) + 2*T(6,3) = 2*720 + 2*120 = 1680.

L. Comtet, Advanced Combinatorics, Reidel, 1974, page 158, exercise 7.

Cf. A000898 (row sums), A001498, A001515, A059343, A060821.

T := (n,k) -> (n!/k!)*binomial(k,n-k)*2^(2*k-n):
seq(seq(T(n, k), k=0..n), n=0..9);

# uses[bell_matrix from A264428]
bell_matrix(lambda n: 2 if n<2 else 0, 12) # Peter Luschny, Jan 19 2016

Defining relation: 2*x*(2*x-1)*...*(2*x-n+1) = sum {k=0..n} T(n, k)*x*(x-1)*...*(x-k+1)
Explicit formula: T(n,k) = (n!/k!)*binomial(k,n-k)*2^(2*k-n). [As defined by Comtet (see reference).]
Recurrence relation: T(n,k) = (2*k-n+1)*T(n-1,k)+2*T(n-1,k-1).
E.g.f.: exp(x*(t^2+2*t)) = 1 + (2*x)*t + (2*x+4*x^2)*t^2/2! + (12*x^2+8*x^3)*t^3/3! + ...
O.g.f. for m-th diagonal (starting at main diagonal m = 0): (2*m)!/m!*x^m/(1-2*x)^(2*m+1).
The triangle is the matrix product [2^k*s(n,k)]n,k>=0 * ([s(n,k)]n,k>=0)^(-1),
where s(n,k) denotes the signed Stirling number of the first kind.
Row sums are [1,2,6,20,76,...] = A000898.
Column sums are [1,4,28,296,...] = [2^n*A001515(n)] n>=0.

A277379 E.g.f.: exp(x/(1-x^2))/sqrt(1-x^2).

Original entry on oeis.org

1, 1, 2, 10, 40, 296, 1936, 17872, 164480, 1820800, 21442816, 279255296, 3967316992, 59837670400, 988024924160, 17009993230336, 318566665977856, 6177885274406912, 129053377688043520, 2786107670662021120, 64136976817284448256, 1525720008470138454016

Offset: 0

Vladimir Reshetnikov, Oct 11 2016

nonn

Is this the same as A227545 (at least for n>=1)?

Eric Weisstein's World of Mathematics, Hermite Polynomial.
Wikipedia, Hermite polynomials.

Cf. A000321, A000898, A059343, A062267, A067994, A227545, A277280, A277281, A277378.

Table[Abs[HermiteH[n, (1 + I)/2]]^2/2^n, {n, 0, 20}]

a(n) = |H_n((1+i)/2)|^2 / 2^n = H_n((1+i)/2) * H_n((1-i)/2) / 2^n, where H_n(x) is n-th Hermite polynomial, i = sqrt(-1).
D-finite with recurrence: (n+1)*(n+2)*(a(n) - n^2*a(n-1)) + (2*n^2+7*n+6)*a(n+1) + a(n+2) = a(n+3).
a(n) ~ n^n * exp(sqrt(2*n)-n) / 2. - Vaclav Kotesovec, Oct 14 2016

cp's OEIS Frontend

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

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A099174 Triangle read by rows: coefficients of modified Hermite polynomials.

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A059344 Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.

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A277281 Maximal coefficient (ignoring signs) in Hermite polynomial of order n.

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A277280 Maximal coefficient in Hermite polynomial of order n.

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A163322 The 3rd Hermite Polynomial evaluated at n: H_3(n) = 8*n^3 - 12*n.

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A163322 The 3rd Hermite Polynomial evaluated at n: H_3(n) = 8n^3 - 12n.