A001464 -id:A001464 - cp's OEIS frontend

A172394 G.f. satisfies: A(x) = G(x/A(x)) where o.g.f. G(x) = A(xG(x)) = Sum_{n>=0} A001464(n)x^n.

1, -1, -1, 0, 1, 0, -4, 0, 27, 0, -248, 0, 2830, 0, -38232, 0, 593859, 0, -10401712, 0, 202601898, 0, -4342263000, 0, 101551822350, 0, -2573779506192, 0, 70282204726396, 0, -2057490936366320, 0, 64291032462761955, 0

Offset: 0

Paul D. Hanna, Feb 06 2010

sign

The e.g.f. of A001464 is exp(-x-x^2/2) = Sum_{n>=0} A001464(n)*x^n/n!.

			G.f.: A(x) = 1 - x - x^2 + x^4 - 4*x^6 + 27*x^8 - 248*x^10 +...
where G(x) = A(x*G(x)) is the o.g.f. of A001464:
G(x) = 1 - x + 2*x^3 - 2*x^4 - 6*x^5 + 16*x^6 + 20*x^7 - 132*x^8 +...
while the e.g.f. of A001464 is given by:
exp(-x-x^2/2) = 1 - x + 2*x^3/3! - 2*x^4/4! - 6*x^5/5! + 16*x^6/6! +...

Cf. A001464, A000699, A172395 (variant).

{a(n)=local(G=sum(m=0,n,m!*polcoeff(exp(-x-x^2/2+x*O(x^m)),m)*x^m)+x*O(x^n));polcoeff(x/serreverse(x*G),n)}

a(2n-2) = (-1)^(n-1)*A000699(n), where A000699(n) is the number of irreducible diagrams with 2n nodes, for n>=1.

a(2n-1) = 0 for n>=2, with a(1) = -1.

A123023 a(n) = (n-1)*a(n-2), a(0)=1, a(1)=0.

Original entry on oeis.org

1, 0, 1, 0, 3, 0, 15, 0, 105, 0, 945, 0, 10395, 0, 135135, 0, 2027025, 0, 34459425, 0, 654729075, 0, 13749310575, 0, 316234143225, 0, 7905853580625, 0, 213458046676875, 0, 6190283353629375, 0, 191898783962510625, 0, 6332659870762850625, 0, 221643095476699771875

Offset: 0

Roger L. Bagula, Sep 24 2006

a(n) is the number of ways of separating n terms into pairs. - Stephen Crowley, Apr 07 2007
a(n) is the n-th moment of the standard normal distribution. - Hal M. Switkay, Nov 06 2019
a(n) is the number of fixed-point free involutions in the symmetric group of degree n. - Nick Krempel, Feb 26 2020

			From _Gus Wiseman_, Dec 23 2018: (Start)
The a(6) = 15 ways of partitioning {1,2,3,4,5,6} into disjoint pairs:
  {{12}{34}{56}},  {{12}{35}{46}},  {{12}{36}{45}},
  {{13}{24}{56}},  {{13}{25}{46}},  {{13}{26}{45}},
  {{14}{23}{56}},  {{14}{25}{36}},  {{14}{26}{35}},
  {{15}{23}{46}},  {{15}{24}{36}},  {{15}{26}{34}},
  {{16}{23}{45}},  {{16}{24}{35}},  {{16}{25}{34}}.
(End)

Richard Bronson, Schaum's Outline of Modern Introductory Differential Equations, MacGraw-Hill, New York, 1973, page 107, solved problem 19.18
Norbert Wiener, Nonlinear Problems in Random Theory, 1958, Equation 1.31

G. C. Greubel, Table of n, a(n) for n = 0..799
Mihai Nica, Terence Tao's central limit theorem, Double Factorials (!!) and the Moment Method, YouTube video (2023).
Sebastian Volz, Design and Implementation of Efficient Algorithms for Operations on Partitions of Sets, Bachelor Thesis, Saarland Univ. (Germany, 2023). See p. 45.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017-2018, Table 2 on p. 15.

Cf. A000085, A000110, A000124, A000142, A000587, A001147, A001464, A006129, A079267, A124794, A186021, A295193.

a:=[1,0]; [n le 2 select a[n] else (n-2)*Self(n-2): n in [1..30]]; // Marius A. Burtea, Nov 07 2019

with(combstruct): ZL2 := [S, {S=Set(Cycle(Z, card=2))}, labeled]:
seq(count(ZL2, size=n), n=0..36); # Zerinvary Lajos, Sep 24 2007
a := n -> ifelse(irem(n, 2) = 1, 0, 2^(n/2) * pochhammer(1/2, n/2)):
seq(a(n), n = 0..36); # Peter Luschny, Jan 11 2023

RecurrenceTable[{a[0] == 1, a[1] == 0, a[n] == (n - 1) a[n - 2]}, a[n], {n, 0, 31}] (* Ray Chandler, Jul 30 2015 *)

a(n) = (1/2)*Gamma((1/2)*n + 1/2)*2^((1/2)*n)*(1 + (-1)^n)/sqrt(Pi). - Stephen Crowley, Apr 07 2007
E.g.f.: exp(x^2/2). - Geoffrey Critzer, Mar 15 2009
a(2n) = A001147(n). - R. J. Mathar, Oct 11 2011
From Sergei N. Gladkovskii, Nov 18 2012, Dec 05 2012, May 16 2013, May 24 2013, Jun 07 2013: (Start)
Continued fractions:
E.g.f.: E(0) where E(k) = 1 + x^2*(4*k+1)/((4*k+2)*(4*k+3) - x^2*(4*k+2)*(4*k+3)^2/(x^2*(4*k+3) + (4*k+4)*(4*k+5)/E(k+1))).
G.f.: 1/G(0) where G(k) =  1 - x^2*(k+1)/G(k+1).
G.f.: 1 + x^2/(1+x) + Q(0)*x^3/(1+x), where Q(k) = 1 + (2*k+3)*x/(1 - x/(x + 1/Q(k+1))).
G.f.: G(0)/2, where G(k) = 1 + 1/(1-x/(x+1/x/(2*k+1)/G(k+1))).
G.f.: (G(0) - 1)*x/(1+x) + 1, where G(k) = 1 + x*(2*k+1)/(1 - x/(x + 1/G(k+1))). (End)
For n even, a(n) = A001147(n/2) = A124794(3^(n/2)). a(n) is also the coefficient of x1*...*xn in Product_{1 <= i < j <= n} (1 + xi*xj). - Gus Wiseman, Dec 23 2018
a(n) = 2^(n/2)*Pochhammer(1/2, n/2)*(n+1 mod 2). - Peter Luschny, Jan 11 2023

Edited by N. J. A. Sloane, Jan 06 2008
Better name by Sergei N. Gladkovskii, May 24 2013
Leading term 1 dropped, offset changed, and entry edited correspondingly by Andrey Zabolotskiy, Nov 07 2019

A066325 Coefficients of unitary Hermite polynomials He_n(x).

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

Christian G. Bower, Dec 14 2001

Also number of involutions on n labeled elements with k fixed points times (-1)^(number of 2-cycles).
Also called normalized Hermite polynomials.
He_n(x) := H_n(x/sqrt(2)) / sqrt(2)^n, with the coefficients of H_n(x) given in A060821. See the Maple program. - Wolfdieter Lang, Jan 13 2020

			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;
    0,  -105,   0,  105,   0, -21,  0, 1;
  ...

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, pp. 89,94 (2.3.41,54).

Robert Israel, Rows n=0..140 of triangle, flattened
P. Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, chapter 8.
P. Diaconis and A. Gamburd, Random matrices, magic squares and matching polynomials, The Electronic Journal of Combinatorics, Volume 11, Issue 2 (2004-6), Research Paper #R2.
E. Elizalde, Cosmology: techniques and observations, arXiv:gr-qc/0409076, 2004.
D. Foata, Une méthode combinatoire pour l'étude des fonctions spéciales, Journées sur les méthodes en mathématiques, Institut Henri Poincaré, Paris 2-3 april 2003.
R. Sazdanovic, A categorification of the polynomial ring, slide presentation, 2011. [_Tom Copeland_, Dec 27 2015]
Index entries for sequences related to Hermite polynomials

Row sums: A001464 (with different signs).
Row sums of absolute values: A000085.
Absolute values are given in A099174.
Cf. A159834, A001147, A060821 (Hermite H_n(x)).

Q:= [seq(orthopoly[H](n,x/sqrt(2))/2^(n/2), n=0..20)]:
seq(seq(coeff(Q[n+1],x,k),k=0..n),n=0..20); # Robert Israel, Jan 01 2016
# Alternative:
T := proc(n,k) option remember; if k > n then 0 elif n = k then 1 else
(T(n, k+2)*(k+2)*(k+1))/(k-n) fi end:
seq(print(seq(T(n, k), k = 0..n)), n = 0..10); # Peter Luschny, Jan 08 2023

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] // Expand; Table[CoefficientList[H[n, x], x], {n, 0, 11}] // Flatten (* Jean-François Alcover, May 11 2015 *)

for(n=0, 12, for(k=0,n, print1(if(Mod(n-k,2)==0, (-2)^((k-n)/2)*n!/(k!*((n-k)/2)!), 0), ", "))) \\ G. C. Greubel, Nov 23 2018

from sympy import Poly
from sympy.abc import x
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 Poly(H(n, x), x).all_coeffs()[::-1]
for n in range(21): print(a(n)) # Indranil Ghosh, May 26 2017

def A066325_row(n):
    T = [0]*(n+1)
    if n==1: return [1]
    for m in (1..n-1):
        a,b,c = 1,0,0
        for k in range(m,-1,-1):
            r = a - (k+1)*c
            if k < m : T[k+2] = u;
            a,b,c = T[k-1],a,b
            u = r
        T[1] = u;
    return T[1:]
for n in (1..11): A066325_row(n)  # Peter Luschny, Nov 01 2012

# uses[riordan_array from A256893]
riordan_array(exp(-x^2/2), x, 8, True) # Peter Luschny, Nov 23 2018

T(n, k) = (-2)^((k-n)/2)*n!/(k!*((n-k)/2)!) for n-k even, 0 otherwise.
E.g.f. of row polynomials {He_n(y)}:  A(x, y) = exp(x*y - x^2/2).
The umbral compositional inverses (cf. A001147) of the polynomials He(n,x) are given by the same polynomials unsigned, A099174. - Tom Copeland, Nov 15 2014
Exp(-D^2/2) x^n = He_n(x) = p_n(x+1) with D = d/dx and p_n(x), the row polynomials of A159834. These are an Appell sequence of polynomials with lowering and raising operators L = D and R = x - D. - Tom Copeland, Jun 26 2018

A144299 Triangle of Bessel numbers read by rows. Row n gives T(n,n), T(n,n-1), T(n,n-2), ..., T(n,0) for n >= 0.

Original entry on oeis.org

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

Offset: 0

David Applegate and N. J. A. Sloane, Dec 06 2008

T(n,k) is the number of partitions of an n-set into k nonempty subsets, each of size at most 2.
The Grosswald and Choi-Smith references give many further properties and formulas.
Considered as an infinite lower triangular matrix T, lim_{n->infinity} T^n = A118930: (1, 1, 2, 4, 13, 41, 166, 652, ...) as a vector. - Gary W. Adamson, Dec 08 2008

			Triangle begins:
  n:
  0: 1
  1: 1  0
  2: 1  1   0
  3: 1  3   0    0
  4: 1  6   3    0   0
  5: 1 10  15    0   0  0
  6: 1 15  45   15   0  0  0
  7: 1 21 105  105   0  0  0  0
  8: 1 28 210  420 105  0  0  0  0
  9: 1 36 378 1260 945  0  0  0  0  0
  ...
The row sums give A000085.
For some purposes it is convenient to rotate the triangle by 45 degrees:
  1 0 0 0 0  0  0   0   0    0    0     0 ...
    1 1 0 0  0  0   0   0    0    0     0 ...
      1 3 3  0  0   0   0    0    0     0 ...
        1 6 15 15   0   0    0    0     0 ...
          1 10 45 105 105    0    0     0 ...
             1 15 105 420  945  945     0 ...
                1  21 210 1260 4725 10395 ...
                    1  28  378 3150 17325 ...
                        1   36  630  6930 ...
                             1   45   990 ...
  ...
The latter triangle is important enough that it has its own entry, A144331. Here the column sums give A000085 and the rows sums give A001515.
If the entries in the rotated triangle are denoted by b1(n,k), n >= 0, k <= 2n, then we have the recurrence b1(n, k) = b1(n - 1, k - 1) + (k - 1)*b1(n - 1, k - 2).
Then b1(n,k) is the number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1 or 2.

E. Grosswald, Bessel Polynomials, Lecture Notes Math., Vol. 698, 1978.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
J. Y. Choi and J. D. H. Smith, On the unimodality and combinatorics of Bessel numbers, Discrete Math., 264 (2003), 45-53.
Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
Toufik Mansour, Matthias Schork, and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
T. Mansour and M. Shattuck, Partial matchings and pattern avoidance, Appl. Anal. Discrete Math. 7 (2013) 25-50.

Other versions of this same triangle are given in A111924 (which omits the first row), A001498 (which left-adjusts the rows in the bottom view), A001497 and A100861. Row sums give A000085.

Cf. A000085, A000217, A001464, A004526, A118930, A144385, A144643..

a144299 n k = a144299_tabl !! n !! k
a144299_row n = a144299_tabl !! n
a144299_tabl = [1] : [1, 0] : f 1 [1] [1, 0] where
   f i us vs = ws : f (i + 1) vs ws where
               ws = (zipWith (+) (0 : map (i *) us) vs) ++ [0]
-- Reinhard Zumkeller, Jan 01 2014

A144299:= func< n,k | k le Floor(n/2) select Factorial(n)/(Factorial(n-2*k)*Factorial(k)*2^k) else 0 >;
[A144299(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 29 2023

Maple code producing the rotated version:
b1 := proc(n, k)
option remember;
if n = k then 1;
elif k < n then 0;
elif n < 1 then 0;
else b1(n - 1, k - 1) + (k - 1)*b1(n - 1, k - 2);
end if;
end proc;
for n from 0 to 12 do lprint([seq(b1(n,k),k=0..2*n)]); od:

T[n_,0]=0; T[1,1]=1; T[2,1]=1; T[n_, k_]:= T[n-1,k-1] + (n-1)T[n-2,k-1];
Table[T[n,k], {n,12}, {k,n,1,-1}]//Flatten (* Robert G. Wilson v *)
Table[If[k<=Floor[n/2],n!/((n-2 k)! k! 2^k),0], {n, 0, 12},{k,0,n}]//Flatten (* Stefano Spezia, Jun 15 2023 *)

def A144299(n,k): return factorial(n)/(factorial(n-2*k)*factorial(k)*2^k) if k <= (n//2) else 0
flatten([[A144299(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 29 2023

T(n, k) = T(n-1, k-1) + (n-1)*T(n-2, k-1).
E.g.f.: Sum_{k >= 0} Sum_{n = 0..2k} T(n,k) y^k x^n/n! = exp(y(x+x^2/2)). (The coefficient of y^k is the e.g.f. for the k-th row of the rotated triangle shown below.)
T(n, k) = n!/((n - 2*k)!*k!*2^k) for 0 <= k <= floor(n/2) and 0 otherwise. - Stefano Spezia, Jun 15 2023
From G. C. Greubel, Sep 29 2023: (Start)
T(n, 1) = A000217(n-1).
Sum_{k=0..n} T(n,k) = A000085(n).
Sum_{k=0..n} (-1)^k*T(n,k) = A001464(n). (End)

Offset fixed by Reinhard Zumkeller, Jan 01 2014

A104556 Matrix inverse of triangle A001497 of Bessel polynomials, read by rows; essentially the same as triangle A096713 of modified Hermite polynomials.

Original entry on oeis.org

1, -1, 1, 0, -3, 1, 0, 3, -6, 1, 0, 0, 15, -10, 1, 0, 0, -15, 45, -15, 1, 0, 0, 0, -105, 105, -21, 1, 0, 0, 0, 105, -420, 210, -28, 1, 0, 0, 0, 0, 945, -1260, 378, -36, 1, 0, 0, 0, 0, -945, 4725, -3150, 630, -45, 1, 0, 0, 0, 0, 0, -10395, 17325, -6930, 990, -55, 1, 0, 0, 0, 0, 0, 10395, -62370, 51975, -13860, 1485, -66, 1

Offset: 0

Paul D. Hanna, Mar 14 2005

Exponential Riordan array [1 - x, x - x^2/2]; cf. A049403. - Peter Bala, Apr 08 2013

Also the Bell transform of (-1)^n if n<2 else 0 and the inverse Bell transform of A001147(n) (adding 1,0,0,... as column 0). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

			Rows begin:
   1;
  -1,  1;
   0, -3,   1;
   0,  3,  -6,    1;
   0,  0,  15,  -10,    1;
   0,  0, -15,   45,  -15,     1;
   0,  0,   0, -105,  105,   -21,     1;
   0,  0,   0,  105, -420,   210,   -28,   1;
   0,  0,   0,    0,  945, -1260,   378, -36,   1;
   0,  0,   0,    0, -945,  4725, -3150, 630, -45, 1; ...
The columns being equal in absolute value to the rows of the matrix inverse A001497:
    1;
    1,   1;
    3,   3,   1;
   15,  15,   6,   1;
  105, 105,  45,  10,  1;
  945, 945, 420, 105, 15, 1; ...

G. C. Greubel, Rows n=0..35 of triangle, flattened
H. Han and S. Seo, Combinatorial proofs of inverse relations and log-concavity for Bessel numbers, Eur. J. Combinat. 29 (7) (2008) 1544-1554. [From _R. J. Mathar_, Mar 20 2009]

Row sums are found in A001464 (offset 1).
Absolute row sums equal A000085.
Cf. A001497, A049403, A096713, A122848, A130757.

With[{nmax = 10}, CoefficientList[CoefficientList[Series[(1 - t)*Exp[x*(t - t^2/2)], {t, 0, nmax}, {x, 0, nmax}], t], x]*Range[0, nmax]!] (* G. C. Greubel, Jun 10 2018 *)

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: (-1)^n if n<2 else 0, 9) # Peter Luschny, Jan 19 2016

E.g.f. : (1 - t)*exp(x*(t - t^2/2)) = 1 + (-1 + x)*t + (-3*x + x^2)*t^2/2! + ... - Peter Bala, Apr 08 2013

A362277 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} (-k/2)^j * binomial(n-j,j)/(n-j)!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -2, 1, 1, 1, -2, -5, -2, 1, 1, 1, -3, -8, 1, 6, 1, 1, 1, -4, -11, 10, 41, 16, 1, 1, 1, -5, -14, 25, 106, 31, -20, 1, 1, 1, -6, -17, 46, 201, -44, -461, -132, 1, 1, 1, -7, -20, 73, 326, -299, -1952, -895, 28, 1, 1, 1, -8, -23, 106, 481, -824, -5123, -1028, 6481, 1216, 1

Offset: 0

Seiichi Manyama, Apr 13 2023

			Square array begins:
  1,  1,  1,   1,    1,    1,     1, ...
  1,  1,  1,   1,    1,    1,     1, ...
  1,  0, -1,  -2,   -3,   -4,    -5, ...
  1, -2, -5,  -8,  -11,  -14,   -17, ...
  1, -2,  1,  10,   25,   46,    73, ...
  1,  6, 41, 106,  201,  326,   481, ...
  1, 16, 31, -44, -299, -824, -1709, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..6 give A000012, (-1)^n * A001464(n), A293604, A362278, A362176, A362279, A362177.
Main diagonal gives A362276.
T(n,2*n) gives A362282.
Cf. A359762, A362302.

T(n,k) = n!*sum(j=0,n\2, (-k/2)^j/(j!*(n-2*j)!));

E.g.f. of column k: exp(x - k*x^2/2).
T(n,k) = T(n-1,k) - k*(n-1)*T(n-2,k) for n > 1.
T(n,k) = n! * Sum_{j=0..floor(n/2)} (-k/2)^j / (j! * (n-2*j)!).

A252284 Exponential generating function exp(-x-x^2-x^3/3).

Original entry on oeis.org

1, -1, -1, 3, 9, -21, -129, 111, 2577, 2871, -57249, -232101, 1175769, 11951523, -6313761, -542318841, -1778088159, 20647593711, 187318128447, -386536525389, -13793029404759, -41926398389541, 783578974052799, 7433562140085663, -22263437361406671, -767083139039850201

Offset: 0

Emanuele Munarini, Dec 16 2014

sign

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A001464, A249015.

S:= series(exp(-x-x^2-x^3/3),x,101):
seq(coeff(S,x,j)*j!,j=0..100); # Robert Israel, Dec 16 2014

a[n_] := Sum[(n!/k!)(-1)^k Sum[Binomial[k,i]Binomial[k-i,n-2i-k]/3^i,{i,0,k}],{k,0,n}]; Table[a[n],{n,0,20}]
With[{nn=30},CoefficientList[Series[Exp[-x-x^2-x^3/3],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jan 01 2021 *)

a(n) := sum((n!/k!)*(-1)^k*sum(binomial(k,i)*binomial(k-i,n-2*i-k)/3^i,i,0,k),k,0,n);
makelist(a(n),n,0,20);

default(seriesprecision, 40); Vec(serlaplace( exp(-x-x^2-x^3/3))) \\ Michel Marcus, Dec 17 2014

a(n) = Sum_{k=0..n} ((n!/k!)*(-1)^k * Sum_{i=0..k} C(k,i)*C(k-i,n-2*i-k)/3^i).
E.g.f.: exp(-x-x^2-x^3/3).
Recurrence: a(n+3)+a(n+2)+2*(n+2)*a(n+1)+(n+2)*(n+1)*a(n)=0.
a(n) = Sum_{k=0..n} 3^k * Stirling1(n,k) * Bell_k(-1/3), where Bell_n(x) is n-th Bell polynomial. - Seiichi Manyama, Jan 31 2024

A334568 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(-Sum_{j=1..k} x^j/j).

Original entry on oeis.org

1, 1, -1, 1, -1, 1, 1, -1, 0, -1, 1, -1, 0, 2, 1, 1, -1, 0, 0, -2, -1, 1, -1, 0, 0, 6, -6, 1, 1, -1, 0, 0, 0, -6, 16, -1, 1, -1, 0, 0, 0, 24, -24, 20, 1, 1, -1, 0, 0, 0, 0, -24, -120, -132, -1, 1, -1, 0, 0, 0, 0, 120, -120, 540, -28, 1, 1, -1, 0, 0, 0, 0, 0, -120, -720, 1764, 1216, -1

Offset: 0

Seiichi Manyama, May 06 2020

			Square array begins:
   1,    1,    1,    1,    1,    1,    1, ...
  -1,   -1,   -1,   -1,   -1,   -1,   -1, ...
   1,    0,    0,    0,    0,    0,    0, ...
  -1,    2,    0,    0,    0,    0,    0, ...
   1,   -2,    6,    0,    0,    0,    0, ...
  -1,   -6,   -6,   24,    0,    0,    0, ...
   1,   16,  -24,  -24,  120,    0,    0, ...
  -1,   20, -120, -120, -120,  720,    0, ...
   1, -132,  540, -720, -720, -720, 5040, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=1..5 give A033999, A001464, A334569, A334570, A334571.

Cf. A330858, A334561.

A(0,k) = 1 and A(n,k) = - (n-1)! * Sum_{j=1..min(k,n)} A(n-j,k)/(n-j)!.

A111882 Row sums of triangle A111595 (normalized rescaled squared Hermite polynomials).

Original entry on oeis.org

1, 1, 0, 4, 4, 36, 256, 400, 17424, 784, 1478656, 876096, 154753600, 560363584, 19057250304, 220388935936, 2564046397696, 83038749753600, 327933273309184, 33173161139160064, 26222822450021376, 14475245839622726656

Offset: 0

Wolfdieter Lang, Aug 23 2005

G. C. Greubel, Table of n, a(n) for n = 0..449

Cf. A111883 (unsigned row sums of A111595).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x/(1+x))/Sqrt(1-x^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jun 10 2018

With[{nmax = 50}, CoefficientList[Series[Exp[x/(1 + x)]/Sqrt[1 - x^2], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Jun 10 2018 *)

x='x+O('x^30); Vec(serlaplace(exp(x/(1+x))/sqrt(1-x^2))) \\ G. C. Greubel, Jun 10 2018

from sympy import hermite, Poly, sqrt
def a(n): return sum(Poly(1/2**n*hermite(n, sqrt(x/2))**2, x).all_coeffs()) # Indranil Ghosh, May 26 2017

E.g.f.: exp(x/(1+x))/sqrt(1-x^2).
a(n) = Sum_{m=0..n} A111595(n, m), n>=0.
A111882(n) = A001464(n)^2. - Mark van Hoeij, Nov 11 2009
D-finite with recurrence a(n) +(n-2)*a(n-1) -(n-1)*(n-2)*a(n-2) -(n-1)*(n-2)^2*a(n-3)=0. - R. J. Mathar, Oct 05 2014

A159834 Coefficient array of Hermite_H(n, (x-1)/sqrt(2))/(sqrt(2))^n.

Original entry on oeis.org

1, -1, 1, 0, -2, 1, 2, 0, -3, 1, -2, 8, 0, -4, 1, -6, -10, 20, 0, -5, 1, 16, -36, -30, 40, 0, -6, 1, 20, 112, -126, -70, 70, 0, -7, 1, -132, 160, 448, -336, -140, 112, 0, -8, 1, -28, -1188, 720, 1344, -756, -252, 168, 0, -9, 1

Offset: 0

Paul Barry, Apr 23 2009

Exponential Riordan array [exp(-x-x^2/2), x].

			Triangle begins:
     1,
    -1,    1,
     0,   -2,    1,
     2,    0,   -3,    1,
    -2,    8,    0,   -4,    1,
    -6,  -10,   20,    0,   -5,    1,
    16,  -36,  -30,   40,    0,   -6,    1,
    20,  112, -126,  -70,   70,    0,   -7,    1,
  -132,  160,  448, -336, -140,  112,    0,   -8,    1
Production matrix is:
  -1,  1,
  -1, -1,  1,
   0, -2, -1,  1,
   0,  0, -3, -1,  1,
   0,  0,  0, -4, -1,  1,
   0,  0,  0,  0, -5, -1,  1,
   0,  0,  0,  0,  0, -6, -1,  1,
   0,  0,  0,  0,  0,  0, -7, -1,  1

G. C. Greubel, Rows n=0..100 of triangle, flattened

Inverse of A111062.
Equal to A066325*(A007318)^{-1}.
First column is A001464.
Row sums are (-1)^n*A001147(n) aerated.
Cf. A133314.

Trow := proc(n) local b, f; b := proc(n, m) option remember; if n < m or m < 0 then
0 elif n = 0 and m = 0 then 1 else b(n-1, m) + b(n-1, m-1) fi end:
f := proc(n) option remember; if n = 0 then 1 elif n = 1 then -1
else f(n-2) - f(n-1) - f(n-2)*n fi end; seq(b(n, k)*f(n-k), k=0..n) end:
seq(Trow(n), n=0..20); # Peter Luschny, Aug 19 2018

T[n_] := CoefficientList[Series[HermiteH[n, (x-1)/Sqrt[2]], {x, 0, 50}], x]/ (Sqrt[2])^n; Table[T[n], {n, 0, 20}] // Flatten (* G. C. Greubel, May 19 2018 *)

row(n) = apply(x->round(x), Vecrev(polhermite(n, (x-1)/sqrt(2))/ (sqrt(2))^n));
tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Aug 11 2018

G.f.: 1/(1-xy+x+x^2/(1-xy+x+2x^2/(1-xy+x+3x^2/(1-xy+x+4x^2/(1-... (continued fraction).
From Tom Copeland, Jun 26 2018: (Start)
E.g.f.: exp[t*p.(x)] = exp[-(t + t^2/2)] e^(x*t).
T(n,k) = binomial(n,k) * A001464(n-k).
These polynomials (p.(x))^n = p_n(x) are an Appell sequence with the lowering and raising operators L = D and R = x - 1 - D, with D = d/dx, such that L p_n(x) = n * p_(n-1)(x) and R p_n(x) = p_(n+1)(x), so the formalism of A133314 applies here, giving recursion relations.
The transpose of the production matrix gives a matrix representation of the raising operator R, with left multiplication of the rows of this entry treated as column vectors.
exp(-(D + D^2/2)) x^n= e^(-D^2/2) (x - 1)^n = He_n(x-1) = p_n(x) = (a. + x)^n, with (a.)^n = a_n = A001464(n) and He_n(x), the unitary or normalized Hermite polynomials of A066325.
A111062 with the e.g.f. exp[t + t^2/2] e^(x*t) gives the matrix inverse for this entry with the umbral inverse polynomials q_n(x), an Appell sequence with the raising operator  x + 1 + D, such that umbrally composed q_n(p.(x)) = x^n = p_n(q.(x)). (End)

cp's OEIS Frontend

A172394 G.f. satisfies: A(x) = G(x/A(x)) where o.g.f. G(x) = A(x*G(x)) = Sum_{n>=0} A001464(n)*x^n.

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A123023 a(n) = (n-1)*a(n-2), a(0)=1, a(1)=0.

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A066325 Coefficients of unitary Hermite polynomials He_n(x).

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A144299 Triangle of Bessel numbers read by rows. Row n gives T(n,n), T(n,n-1), T(n,n-2), ..., T(n,0) for n >= 0.

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A104556 Matrix inverse of triangle A001497 of Bessel polynomials, read by rows; essentially the same as triangle A096713 of modified Hermite polynomials.

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A362277 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} (-k/2)^j * binomial(n-j,j)/(n-j)!.

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A172394 G.f. satisfies: A(x) = G(x/A(x)) where o.g.f. G(x) = A(xG(x)) = Sum_{n>=0} A001464(n)x^n.