A001814 -id:A001814 - 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

A001879 a(n) = (2n+2)!/(n!*2^(n+1)).

Original entry on oeis.org

1, 6, 45, 420, 4725, 62370, 945945, 16216200, 310134825, 6547290750, 151242416325, 3794809718700, 102776096548125, 2988412653476250, 92854250304440625, 3070380543400170000, 107655217802968460625, 3989575718580595893750, 155815096120119939628125

Offset: 0

N. J. A. Sloane

From Wolfdieter Lang, Oct 06 2008: (Start)
a(n) is the denominator of the n-th approximant to the continued fraction 1^2/(6+3^2/(6+5^2/(6+... for Pi-3. W. Lang, Oct 06 2008, after an e-mail from R. Rosenthal. Cf. A142970 for the corresponding numerators.
The e.g.f. g(x)=(1+x)/(1-2*x)^(5/2) satisfies (1-4*x^2)*g''(x) - 2*(8*x+3)*g'(x) -9*g(x) = 0 (from the three term recurrence given below). Also g(x)=hypergeom([2,3/2],[1],2*x). (End)
Number of descents in all fixed-point-free involutions of {1,2,...,2(n+1)}. A descent of a permutation p is a position i such that p(i) > p(i+1). Example: a(1)=6 because the fixed-point-free involutions 2143, 3412, and 4321 have 2, 1, and 3 descents, respectively. - Emeric Deutsch, Jun 05 2009
First differences of A193651. - Vladimir Reshetnikov, Apr 25 2016
a(n-2) is the number of maximal elements in the absolute order of the Coxeter group of type D_n. - Jose Bastidas, Nov 01 2021

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77 (Problem 10, values of Bessel polynomials).
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).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Reinis Cirpons, James East, and James D. Mitchell, Transformation representations of diagram monoids, arXiv:2411.14693 [math.RA], 2024. See pp. 3, 33.
Selden Crary, Richard Diehl Martinez and Michael Saunders, The Nu Class of Low-Degree-Truncated Rational Multifunctions. Ib. Integrals of Matern-correlation functions for all odd-half-integer class parameters, arXiv:1707.00705 [stat.ME], 2017, Table 1.
Alexander Kreinin, Integer Sequences and Laplace Continued Fraction, Preprint 2016.
J. Riordan, Notes to N. J. A. Sloane, Jul. 1968

Cf. A002544, A001814, A001876, A001877, A001878.
Second column of triangle A001497. Equals (A001147(n+1)-A001147(n))/2.
Equals row sums of A163938.

[Factorial(2*n+2)/(Factorial(n)*2^(n+1)): n in [0..20]]; // Vincenzo Librandi, Nov 22 2011

restart: G(x):=(1-x)/(1-2*x)^(1/2): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od:x:=0:seq(f[n],n=2..20); # Zerinvary Lajos, Apr 04 2009

Table[(2n+2)!/(n!2^(n+1)),{n,0,20}] (* Vincenzo Librandi, Nov 22 2011 *)

PARI
```
a(n)=if(n<0,0,(2*n+2)!/n!/2^(n+1))
```

E.g.f.: (1+x)/(1-2*x)^(5/2).
a(n)*n = a(n-1)*(2n+1)*(n+1); a(n) = a(n-1)*(2n+4)-a(n-2)*(2n-1), if n>0. - Michael Somos, Feb 25 2004
From Wolfdieter Lang, Oct 06 2008: (Start)
a(n) = (n+1)*(2*n+1)!! with the double factorials (2*n+1)!!=A001147(n+1).
D-finite with recurrence a(n) = 6*a(n-1) + ((2*n-1)^2)*a(n-2), a(-1)=0, a(0)=1. (End)
With interpolated 0's, e.g.f.: B(A(x)) where B(x)= x exp(x) and A(x)=x^2/2.
E.g.f.: -G(0)/2 where G(k) = 1 - (2*k+3)/(1 - x/(x - (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 06 2012
G.f.: (1-x)/(2*x^2*Q(0)) - 1/(2*x^2), where Q(k) = 1 - x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 20 2013
From Karol A. Penson, Jul 12 2013: (Start)
Integral representation as n-th moment of a signed function w(x) of bounded variation on (0,infinity),
  w(x) = -(1/4)*sqrt(2)*sqrt(x)*(1-x)*exp(-x/2)/sqrt(Pi):
  a(n) = Integral_{x>=0} x^n*w(x), n>=0.
  For x>1, w(x)>0. w(0)=w(1)=limit(w(x),x=infinity)=0. For x<1, w(x)<0.
Asymptotics: a(n)->(1/576)*2^(1/2+n)*(1152*n^2+1680*n+505)*exp(-n)*(n)^(n), for n->infinity. (End)
G.f.: 2F0(3/2,2;;2x). - R. J. Mathar, Aug 08 2015

Entry revised Aug 31 2004 (thanks to Ralf Stephan and Michael Somos)

E.g.f. in comment line corrected by Wolfdieter Lang, Nov 21 2011

A126804 a(n) = (2n)! / (n-1)!.

Original entry on oeis.org

2, 24, 360, 6720, 151200, 3991680, 121080960, 4151347200, 158789030400, 6704425728000, 309744468633600, 15543540607795200, 841941782922240000, 48962152914554880000, 3042648073975910400000, 201220459292273541120000, 14110584707870682071040000

Offset: 1

Jonathan R. Love (japanada11(AT)yahoo.ca), Feb 22 2007

Old name was "Multiplying n X n integers above n".
a(n) = 2*A001814(n). - Zerinvary Lajos, May 03 2007
A179214(n) <= a(n). - Reinhard Zumkeller, Jul 05 2010
Product of the numbers from n to 2n. - Wesley Ivan Hurt, Dec 14 2015

			a(5) = 151200 because five digits above 5: (6, 7, 8, 9, 10), multiplied by five equals 5*(6*7*8*9*10) = 151200.

Robert Israel, Table of n, a(n) for n = 1..360
Elliot J. Carr and Matthew J. Simpson, Accurate and efficient calculation of finite response times for groundwater flow, arXiv:1707.06331 [physics.flu-dyn], 2017, p. 11.

Cf. A045943, A073838. - Reinhard Zumkeller, Jul 05 2010

Cf. A001814, A179214.

[Factorial(2*n)/Factorial(n-1) : n in [1..20]]; // Wesley Ivan Hurt, Dec 14 2015

a:=n->sum((count(Permutation(2*n+2),size=n+1)),j=0..n): seq(a(n), n=0..15); # Zerinvary Lajos, May 03 2007
seq(mul((n+k), k=0..n), n=1..16); # Zerinvary Lajos, Sep 21 2007
with(combstruct):with(combinat) :bin := {B=Union(Z,Prod(B,B))}: seq (count([B,bin,labeled],size=n)*(n-1), n=2..17); # Zerinvary Lajos, Dec 05 2007

Table[Pochhammer[n, n + 1], {n, 17}] (* Arkadiusz Wesolowski, Aug 13 2012 *)
Table[(2 n)!/(n - 1)!, {n, 20}] (* Wesley Ivan Hurt, Dec 14 2015 *)

a(n) = prod(k=n, 2*n, k); \\ Michel Marcus, Dec 15 2015

x='x+O('x^99); Vec(serlaplace(2*x/(1-4*x)^(3/2))) \\ Altug Alkan, Mar 11 2018

a(n) = (2n)! / (n-1)!.
a(n) = Product_{i=n..2n} i. - Wesley Ivan Hurt, Dec 14 2015
From Robert Israel, Dec 15 2015: (Start)
a(n) = (2*n*(2*n-1)/(n-1))*a(n-1).
E.g.f.: 2*x/(1-4*x)^(3/2). (End)
a(n) = Pochhammer(n,n+1). - Pedro Caceres, Mar 10 2018

New name from Wesley Ivan Hurt, Dec 15 2015

A067147 Triangle of coefficients for expressing x^n in terms of Hermite polynomials.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 6, 0, 1, 12, 0, 12, 0, 1, 0, 60, 0, 20, 0, 1, 120, 0, 180, 0, 30, 0, 1, 0, 840, 0, 420, 0, 42, 0, 1, 1680, 0, 3360, 0, 840, 0, 56, 0, 1, 0, 15120, 0, 10080, 0, 1512, 0, 72, 0, 1, 30240, 0, 75600, 0, 25200, 0, 2520, 0, 90, 0, 1

Offset: 0

Christian G. Bower, Jan 03 2002

x^n = (1/2^n) * Sum_{k=0..n} a(n,k)*H_k(x).

These polynomials, H_n(x), are an Appell sequence, whose umbral compositional inverse sequence HI_n(x) consists of the same polynomials signed with the e.g.f. e^{-t^2} e^{xt}. Consequently, under umbral composition H_n(HI.(x)) = x^n = HI_n(H.(x)). Other differently scaled families of Hermite polynomials are A066325, A099174, and A060821. See Griffin et al. for a relation to the Catalan numbers and matrix integration. - Tom Copeland, Dec 27 2020

			Triangle begins with:
    1;
    0,   1;
    2,   0,   1;
    0,   6,   0,   1;
   12,   0,  12,   0,   1;
    0,  60,   0,  20,   0,   1;
  120,   0, 180,   0,  30,   0,   1;

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. (Table 22.12)

G. C. Greubel, 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 [alternative scanned copy].
M. Griffin, K. Ono, L. Rolen, and D. Zagier, Jensen polynomials for the Riemann zeta function and other sequences, arXiv:1902.07321 [math.NT], 2019.
Index entries for sequences related to Hermite polynomials

Row sums give A047974. Columns 0-2: A001813, A000407, A001814. Cf. A048854, A060821.

Cf. A060821, A066325, and A099174.

[[Round(Factorial(n)*(1+(-1)^(n+k))/(2*Factorial(k)*Gamma((n-k+2)/2))): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Jun 09 2018

T := proc(n, k) (n - k)/2; `if`(%::integer, (n!/k!)/%!, 0) end:
for n from 0 to 11 do seq(T(n, k), k=0..n) od; # Peter Luschny, Jan 05 2021

Table[n!*(1+(-1)^(n+k))/(2*k!*Gamma[(n-k+2)/2]), {n,0,20}, {k,0,n}]// Flatten (* G. C. Greubel, Jun 09 2018 *)

T(n, k) = round(n!*(1+(-1)^(n+k))/(2*k! *gamma((n-k+2)/2)))
for(n=0,20, for(k=0,n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jun 09 2018

{T(n,k) = if(k<0 || nMichael Somos, Jan 15 2020 */

E.g.f. (rel to x): A(x, y) = exp(x*y + x^2).
Sum_{ k>=0 } 2^k*k!*T(m, k)*T(n, k) = T(m+n, 0) = |A067994(m+n)|. - Philippe Deléham, Jul 02 2005
T(n, k) = 0 if n-k is odd; T(n, k) = n!/(k!*((n-k)/2)!) if n-k is even. - Philippe Deléham, Jul 02 2005
T(n, k) = n!/(k!*2^((n-k)/2)*((n-k)/2)!)*2^((n+k)/2)*(1+(-1)^(n+k))/2^(k+1).
T(n, k) = A001498((n+k)/2, (n-k)/2)2^((n+k)/2)(1+(-1)^(n+k))/2^(k+1). - Paul Barry, Aug 28 2005
Exponential Riordan array (e^(x^2),x). - Paul Barry, Sep 12 2006
G.f.: 1/(1-x*y-2*x^2/(1-x*y-4*x^2/(1-x*y-6*x^2/(1-x*y-8*x^2/(1-... (continued fraction). - Paul Barry, Apr 10 2009
The n-th row entries may be obtained from D^n(exp(x*t)) evaluated at x = 0, where D is the operator sqrt(1+4*x)*d/dx. - Peter Bala, Dec 07 2011
As noted in the comments this is an Appell sequence of polynomials, so the lowering and raising operators defined by L H_n(x) = n H_{n-1}(x) and R H_{n}(x) = H_{n+1}(x) are L = D_x, the derivative, and R = D_t log[e^{t^2} e^{xt}] |{t = D_x} = x + 2 D_x, and the polynomials may also be generated by e^{-D^2} x^n = H_n(x). - _Tom Copeland, Dec 27 2020

A119836 Bi-diagonal inverse of [k<=n]*n!/(2k)!.

Original entry on oeis.org

1, -2, 2, 0, -24, 12, 0, 0, -360, 120, 0, 0, 0, -6720, 1680, 0, 0, 0, 0, -151200, 30240, 0, 0, 0, 0, 0, -3991680, 665280, 0, 0, 0, 0, 0, 0, -121080960, 17297280, 0, 0, 0, 0, 0, 0, 0, -4151347200, 518918400, 0, 0, 0, 0, 0, 0, 0, 0, -158789030400, 17643225600, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6704425728000, 670442572800

Offset: 0

Paul Barry, May 25 2006

Row sums are A119837. T(n,n)=A001813(n). T(n,n-1)=2*A001814(n)

			Triangle begins
1,
-2, 2,
0, -24, 12,
0, 0, -360, 120,
0, 0, 0, -6720, 1680,
0, 0, 0, 0, -151200, 30240,
0, 0, 0, 0, 0, -3991680, 665280,
0, 0, 0, 0, 0, 0, -121080960, 17297280,
0, 0, 0, 0, 0, 0, 0, -4151347200, 518918400,
0, 0, 0, 0, 0, 0, 0, 0, -158789030400, 17643225600,
0, 0, 0, 0, 0, 0, 0, 0, 0, -6704425728000, 670442572800

Triangle T(n,n)=(2n)!/n!,T(n,n-1)=(2n)!/(n-1)!,T(n,k)=0 otherwise.

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Python

Formula

A001879 a(n) = (2n+2)!/(n!*2^(n+1)).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A126804 a(n) = (2n)! / (n-1)!.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

Extensions

A067147 Triangle of coefficients for expressing x^n in terms of Hermite polynomials.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

A119836 Bi-diagonal inverse of [k<=n]*n!/(2k)!.

Views

Author

Keywords

Comments

Examples

Formula

A119837 a(n)=(2n)!/n!-(2n)!/(n-1)!.

Views

Author

Keywords

Comments

Programs

Mathematica

Formula