A062267 -id:A062267 - 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

A000321 H_n(-1/2), where H_n(x) is Hermite polynomial of degree n.

Original entry on oeis.org

1, -1, -1, 5, 1, -41, 31, 461, -895, -6481, 22591, 107029, -604031, -1964665, 17669471, 37341149, -567425279, -627491489, 19919950975, 2669742629, -759627879679, 652838174519, 31251532771999, -59976412450835, -1377594095061119, 4256461892701199, 64623242860354751

Offset: 0

N. J. A. Sloane

Binomial transform gives A067994. Inverse binomial transform gives A062267(n)*(-1)^n. - Vladimir Reshetnikov, Oct 11 2016

The congruence a(n+k) == (-1)^k*a(n) (mod k) holds for all n and k. It follows that for even k the sequence obtained by reducing a(n) modulo k is purely periodic with period a divisor of k, while for odd k the sequence obtained by reducing a(n) modulo k is purely periodic with period a divisor of 2*k. See A047974. - Peter Bala, Apr 10 2023

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 209.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..732 (terms 0..200 from T. D. Noe)
Koichi, Yamamoto, An asymptotic series for the number of three-line Latin rectangles, J. Math. Soc. Japan 1, (1950). 226-241.
Index entries for sequences related to Hermite polynomials

Cf. A000186, A062267, A144141, A293604.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(-x-x^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jun 09 2018

Table[HermiteH[n, -1/2], {n, 0, 25}] (* Vladimir Joseph Stephan Orlovsky, Jun 15 2009 *)
Table[(-2)^n HypergeometricU[-n/2, 1/2, 1/4], {n, 0, 25}] (* Benedict W. J. Irwin, Oct 17 2017 *)

N=66;  x='x+O('x^N);
egf=exp(-x-x^2);  Vec(serlaplace(egf))
/* Joerg Arndt, Mar 07 2013 */

vector(50, n, n--; sum(k=0, n/2, (-1)^(n-k)*k!*binomial(n, k)*binomial(n-k, k))) \\ Altug Alkan, Oct 22 2015

a(n) = polhermite(n, -1/2); \\ Michel Marcus, Oct 12 2016

from sympy import hermite
def a(n): return hermite(n, -1/2) # Indranil Ghosh, May 26 2017

E.g.f.: exp(-x-x^2).
a(n) = Sum_{k=0..floor(n/2)} (-1)^(n-k)*k!*C(n, k)*C(n-k, k).
a(n) = - a(n-1) - 2*(n-1)*a(n-2), a(0) = 1, a(1) = -1.
A000186(n) ~ n!^2*exp(1)^(-3)*(a(0) + a(1)/n + a(2)/(2*[n]2) + ... + a(k)/(k!*[n]_k) + ...), where [n]_k = n*(n-1)*...*(n-k + 1), [n]_0 = 1. - _Vladeta Jovovic, Apr 30 2001
a(n) = Sum_{k=0..n} (-1)^(2*n-k)*C(k,n-k)*n!/k!. - Paul Barry, Oct 08 2007, corrected by Altug Alkan, Oct 22 2015
E.g.f.: 1 - x*(1 - E(0) )/(1+x) where E(k) = 1 - (1+x)/(k+1)/(1-x/(x+1/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 18 2013
E.g.f.: -x/Q(0) where Q(k) = 1 - (1+x)/(1 - x/(x - (k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 06 2013
G.f.: 1/(x*Q(0)), where Q(k) = 1 + 1/x + 2*(k+1)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 21 2013
a(n) = (-2)^n * U(-n/2, 1/2, 1/4), where U is the confluent hypergeometric function. - Benedict W. J. Irwin, Oct 17 2017
E.g.f.: Product_{k>=1} (1 + (-x)^k)^(mu(k)/k). - Ilya Gutkovskiy, May 26 2019

Formulae and more terms from Vladeta Jovovic, Apr 30 2001

A122021 a(n) = a(n-2) - (n-1)*a(n-3), with a(0) = 0, a(1) = 1, a(2) = 2.

Original entry on oeis.org

0, 1, 2, 1, -1, -7, -6, -1, 43, 47, 52, -383, -465, -1007, 4514, 5503, 19619, -66721, -73932, -419863, 1193767, 1058777, 10010890, -25204097, -14340981, -265465457, 615761444, 107400049, 7783328783, -17133920383, 4668727362

Offset: 0

Roger L. Bagula, Sep 12 2006

sign

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000898, A121966, A062267.

a:= function(n)
    if n<3 then return n;
    else return a(n-2) - (n-1)*a(n-3);
    fi;
  end;
List([0..30], n-> a(n) ); # G. C. Greubel, Oct 06 2019

I:=[0,1,2]; [n le 3 select I[n] else Self(n-2) - (n-2)*Self(n-3): n in [1..30]]; // G. C. Greubel, Oct 06 2019

a:= proc(n) option remember;
if n < 3 then n
else a(n-2)-(n-1)*a(n-3)
fi;
end proc;
seq(a(n), n = 0..30); # G. C. Greubel, Oct 06 2019

a[0]=0; a[1]=1; a[2]=2; a[n_]:= a[n]= a[n-2] - (n-1)*a[n-3]; Table[a[n], {n, 0, 30}]
RecurrenceTable[{a[0]==0,a[1]==1,a[2]==2,a[n]==a[n-2]-(n-1)a[n-3]},a,{n,30}] (* Harvey P. Dale, Apr 29 2022 *)

my(m=30, v=concat([0,1,2], vector(m-3))); for(n=4, m, v[n] = v[n-2] - (n-2)*v[n-3]); v \\ G. C. Greubel, Oct 06 2019

def a(n):
    if (n<3): return n
    else: return a(n-2) - (n-1)* a(n-3)
[a(n) for n in (0..30)] # G. C. Greubel, Oct 06 2019

Edited by N. J. A. Sloane, Sep 12 2006

Offset changed by G. C. Greubel, Oct 06 2019

A122022 a(n) = a(n-1) - (n-1)*a(n-4), with a(0) = 0, a(1) = 1, a(2) = 2, a(3) = 1.

Original entry on oeis.org

0, 1, 2, 1, 1, -3, -13, -19, -26, -2, 115, 305, 591, 615, -880, -5150, -14015, -23855, -8895, 83805, 350090, 827190, 1013985, -829725, -8881795, -28734355, -54083980, -32511130, 207297335, 1011859275, 2580294695, 3555628595, -2870588790, -35250085590

Offset: 0

Roger L. Bagula, Sep 12 2006

sign

Harvey P. Dale, Table of n, a(n) for n = 0..999 [Offset adapted by _Georg Fischer_, Jun 06 2021]

Cf. A000898, A062267, A121966, A122050.

a:= function(n)
    if n<3 then return n;
    elif n=3 then return 1;
    else return a(n-1) - (n-1)*a(n-4);
    fi;
  end;
List([1..30], n-> a(n) ); # G. C. Greubel, Oct 06 2019

I:=[0,1,2,1]; [n le 4 select I[n] else Self(n-1) - (n-2)*Self(n-4): n in [1..30]]; // G. C. Greubel, Oct 06 2019

a:= proc(n) option remember;
      if n<3 then n
    elif n=3 then 1
    else a(n-1) - (n-1)*a(n-4)
      fi;
end: seq(a(n), n=0..30); # G. C. Greubel, Oct 06 2019

a[0]=0; a[1]=1; a[2]=2; a[3]=1; a[n_]:= a[n]= a[n-1] - (n-1)*a[n-4]; Table[a[n], {n, 0, 30}]
RecurrenceTable[{a[0]==0,a[1]==1,a[2]==2,a[3]==1,a[n]==a[n-1]- (n-1) a[n-4]},a,{n,0,30}] (* Harvey P. Dale, Nov 28 2014 *)

my(m=30, v=concat([0,1,2,1], vector(m-4))); for(n=5, m, v[n] = v[n-1] - (n-2)*v[n-4]); v \\ G. C. Greubel, Oct 06 2019

@CachedFunction
def a(n):
    if (n<3): return n
    elif (n==3): return 1
    else: return a(n-1) - (n-1)*a(n-4)
[a(n) for n in (0..30)] # G. C. Greubel, Oct 06 2019

Edited by N. J. A. Sloane, Sep 12 2006

Offset corrected by Georg Fischer, Jun 06 2021

A369738 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(1 - (1+x)^k).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -3, 2, -1, 0, 1, -4, 3, 4, 1, 0, 1, -5, 4, 21, -20, -1, 0, 1, -6, 5, 56, -63, 8, 1, 0, 1, -7, 6, 115, -104, -423, 184, -1, 0, 1, -8, 7, 204, -95, -2464, 1899, -464, 1, 0, 1, -9, 8, 329, 36, -8245, 1696, 15201, -1648, -1, 0

Offset: 0

Seiichi Manyama, Jan 30 2024

			Square array T(n,k) begins:
  1,  1,   1,    1,     1,      1,       1, ...
  0, -1,  -2,   -3,    -4,     -5,      -6, ...
  0,  1,   2,    3,     4,      5,       6, ...
  0, -1,   4,   21,    56,    115,     204, ...
  0,  1, -20,  -63,  -104,    -95,      36, ...
  0, -1,   8, -423, -2464,  -8245,  -21096, ...
  0,  1, 184, 1899,  1696, -21275, -124344, ...

Columns k=0..5 give A000007, A033999, (-1)^n * A062267(n), A369751, A369752, A369753.
Main diagonal gives A369754.
Cf. A000587, A294042.

a000587(n) = sum(k=0, n, (-1)^k*stirling(n, k, 2));
T(n, k) = sum(j=0, n, k^j*stirling(n, j, 1)*a000587(j));

T(0,k) = 1; T(n,k) = -k * (n-1)! * Sum_{j=1..min(k,n)} binomial(k-1,j-1) * T(n-j,k)/(n-j)!.

T(n,k) = Sum_{j=0..n} k^j * Stirling1(n,j) * A000587(j).

A144141 a(n) = Hermite(n,2).

Original entry on oeis.org

1, 4, 14, 40, 76, -16, -824, -3104, -880, 46144, 200416, -121216, -4894016, -16666880, 60576896, 708980224, 1018614016, -18612911104, -109084520960, 233726715904, 5080118660096, 10971406004224, -169479359707136, -1160659303014400, 3153413334470656

Offset: 0

Vladimir Joseph Stephan Orlovsky, Sep 11 2008

sign

G. C. Greubel, Table of n, a(n) for n = 0..729

Cf. A000898, A000321, A062267.

[(&+[(-1)^k*Factorial(n)*(4)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Jul 10 2018

lst={};Do[AppendTo[lst,HermiteH[n,2]],{n,0,7^2}];lst
HermiteH[Range[0,30],2]  (* Harvey P. Dale, May 20 2012 *)

for(n=0, 50, print1(polhermite(n, 2), ", " )) \\ G. C. Greubel, Jul 10 2018

From G. C. Greubel, Jul 10 2018: (Start)
E.g.f.: exp(4*x - x^2).
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*n!*4^(n-2*k)/(k!*(n-2*k)!). (End)

A122033 a(n) = 2a(n-1) - 2(n-2)*a(n-2), with a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 4, 4, -8, -40, -16, 368, 928, -3296, -21440, 16448, 461696, 561536, -9957632, -34515200, 209783296, 1455022592, -3803020288, -57076808704, 22755112960, 2214428956672, 3518653394944, -85968709390336, -326758168158208, 3301044295639040, 22286480662872064

Offset: 0

Roger L. Bagula, Sep 13 2006

G. C. Greubel, Table of n, a(n) for n = 0..725

Cf. A000898, A121966, A328141.

a:=[1,2];; for n in [3..35] do a[n]:=2*(a[n-1]-(n-3)*a[n-2]); od; a; # G. C. Greubel, Oct 04 2019

I:=[1,2]; [n le 2 select I[n] else 2*(Self(n-1)-(n-3)*Self(n-2)): n in [1..35]]; // G. C. Greubel, Oct 04 2019

a:= proc(n) option remember;
      if n < 2 then n+1
    else 2*(a(n-1) - (n-2)*a(n-2))
      fi
    end proc:
seq(a(n), n = 0..35); # G. C. Greubel, Oct 04 2019

a[n_]:= a[n]= If[n<2, n+1, a[n-1]-(n-2)*a[n-2]]; Table[a[n], {n,0,30}] (* modified by G. C. Greubel, Oct 04 2019 *)
nxt[{n_,a_,b_}]:={n+1,b,2b-2a(n-1)}; NestList[nxt,{1,1,2},30][[;;,2]] (* Harvey P. Dale, Jan 01 2024 *)

my(m=35, v=concat([1,2], vector(m-2))); for(n=3, m, v[n] = 2*(v[n-1] - (n-3)*v[n-2] ) ); v \\ G. C. Greubel, Oct 04 2019

def a(n):
    if n<2: return n+1
    else: return 2*(a(n-1) - (n-2)*a(n-2))
[a(n) for n in (0..35)] # G. C. Greubel, Oct 04 2019

a(n) = 2*a(n-1) - 2*(n-2)*a(n-2), with a(0)=1, a(1)=2. - G. C. Greubel, Oct 04 2019
a(n) = 2*A062267(n-1) for n > 0. - Michel Marcus, Oct 05 2019
E.g.f.: 1 + exp(1)*sqrt(Pi)*( erf(1) - erf(1-x) ), where erf(x) is the error function. - G. C. Greubel, Oct 05 2019

Edited by N. J. A. Sloane, Sep 17 2006

Corrected and offset changed by G. C. Greubel, Oct 04 2019

A181089 Triangle T(n, k) = A060821(n,k) + A060821(n,n-k), read by rows.

Original entry on oeis.org

2, 2, 2, 2, 0, 2, 8, -12, -12, 8, 28, 0, -96, 0, 28, 32, 120, -160, -160, 120, 32, -56, 0, 240, 0, 240, 0, -56, 128, -1680, -1344, 3360, 3360, -1344, -1680, 128, 1936, 0, -17024, 0, 26880, 0, -17024, 0, 1936, 512, 30240, -9216, -80640, 48384, 48384, -80640, -9216, 30240, 512

Offset: 0

Roger L. Bagula, Oct 02 2010

			Triangle begins as:
     2;
     2,     2;
     2,     0,      2;
     8,   -12,    -12,      8;
    28,     0,    -96,      0,      28;
    32,   120,   -160,   -160,     120,    32;
   -56,     0,    240,      0,     240,     0,     -56;
   128, -1680,  -1344,   3360,    3360, -1344,   -1680,   128;
  1936,     0, -17024,      0,   26880,     0,  -17024,     0,   1936;
   512, 30240,  -9216, -80640,   48384, 48384,  -80640, -9216,  30240, 512;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A060821, A062267.

(* First program *)
p[x_, n_] = HermiteH[n, x] + ExpandAll[x^n*HermiteH[n, 1/x]];
Flatten[Table[CoefficientList[p[x, n], x], {n, 0, 15}]] (* edited by G. C. Greubel, Apr 04 2021 *)
(* Second program *)
A060821[n_, k_]:= If[EvenQ[n-k], (-1)^(Floor[(n-k)/2])*2^k*n!/(k!*(Floor[(n - k)/2]!)), 0];
T[n_, k_]:= A060821[n, k] +A060821[n, n-k];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 04 2021 *)

def A060821(n,k): return (-1)^((n-k)//2)*2^k*factorial(n)/(factorial(k)*factorial( (n-k)//2)) if (n-k)%2==0 else 0
def T(n,k): return A060821(n, k) + A060821(n, n-k)
flatten([[T(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Apr 04 2021

T(n, k) = coefficients [x^k] of the polynomial HermiteH(n,x) + x^n*HermiteH(n,1/x).
T(n, k) = A060821(n,k) + A060821(n,n-k).
Sum_{k=0..n} T(n, k) = 2*A062267(n).

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

A025165 a(n) = H_n(1) / 2^floor(n/2) where H_n is the n-th Hermite polynomial.

Original entry on oeis.org

1, 2, 1, -2, -5, -2, 23, 58, -103, -670, 257, 7214, 4387, -77794, -134825, 819466, 2841841, -7427774, -55739071, 22221790, 1081264139, 1718092478, -20988454441, -79774943398, 402959508745

Offset: 0

Wouter Meeussen

sign

G. C. Greubel, Table of n, a(n) for n = 0..800
Index entries for sequences related to Hermite polynomials

[((&+[(-1)^k*Factorial(n)*(2)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]]))/2^(Floor(n/2)): n in [0..30]]; // G. C. Greubel, Jul 10 2018

A025165 := proc(n)
    HermiteH(n,1)/2^(floor(n/2)) ;
    simplify(%) ;
end proc: # R. J. Mathar, Feb 05 2013

Table[ HermiteH[ n, 1 ]/2^Floor[ n/2 ], {n, 0, 24} ]

for(n=0,30, print1(polhermite(n,1)/2^(floor(n/2)), ", ")) \\ G. C. Greubel, Jul 10 2018

a(n) = A062267(n)/A016116(n). - R. J. Mathar, Feb 05 2013

Conjecture: a(n) +a(n-1) +(2*n-5)*a(n-2) +(2*n-7)*a(n-3) +(n-2)*(n-3)*a(n-4) +(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Feb 25 2015

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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A000321 H_n(-1/2), where H_n(x) is Hermite polynomial of degree n.

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A122021 a(n) = a(n-2) - (n-1)*a(n-3), with a(0) = 0, a(1) = 1, a(2) = 2.

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A122022 a(n) = a(n-1) - (n-1)*a(n-4), with a(0) = 0, a(1) = 1, a(2) = 2, a(3) = 1.

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A369738 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(1 - (1+x)^k).

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A144141 a(n) = Hermite(n,2).

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A122033 a(n) = 2a(n-1) - 2(n-2)*a(n-2), with a(0)=1, a(1)=2.