A052127 -id:A052127 - cp's OEIS frontend

A052124 Expansion of e.g.f. exp(-2*x)/(1-x)^3.

1, 1, 4, 16, 88, 568, 4288, 36832, 354688, 3781504, 44199424, 561823744, 7714272256, 113769309184, 1793341407232, 30085661765632, 535170830467072, 10060645294440448, 199287423535808512, 4148644277780217856, 90545807649965080576, 2067407731760475406336, 49285894020028992323584

Offset: 0

N. J. A. Sloane, Jan 23 2000

nonn

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.64(b).

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A052127, A082031.
Cf. A000023, A087981.
Cf. A000153, A137775.

A052124 := proc(n) option remember; if n <=1 then 1 else n*A052124(n-1)+2*(n-1)*A052124(n-2); fi; end; # Detlef Pauly

Table[(n+5)*(n+2)*n!*Sum[(-1)^k*2^(k+2)*(k+3)/(k+5)!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 28 2012 *)
With[{nn=20},CoefficientList[Series[Exp[(-2x)]/(1-x)^3,{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Oct 23 2017 *)

my(x='x+O('x^25)); Vec(serlaplace( exp(-2*x)/(1-x)^3)) \\ Michel Marcus, Oct 25 2021

from math import factorial
from fractions import Fraction
def A052124(n): return int((n+5)*(n+2)*factorial(n)*sum(Fraction((-1 if k&1 else 1)*(k+3)<Chai Wah Wu, Apr 20 2023

a(n) = n*a(n-1) + 2*(n-1)*a(n-2). - Detlef Pauly (dettodet(AT)yahoo.de), Sep 22 2003
a(n) = (n+5)*(n+2)*n! * Sum_{k=0..n} (-1)^k*2^(k+2)*(k+3)/(k+5)!. - Vaclav Kotesovec, Oct 28 2012
G.f.: 1/Q(0), where Q(k) = 1 + 2*x - x*(k+3)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013
a(n) ~ n!*(n+5)*(n+2)/(2*exp(2)). - Vaclav Kotesovec, Jun 15 2013
From Peter Bala, Sep 20 2013: (Start)
a(n) ~ (1/2)*n^2*n!/e^2 for large n.
Letting n -> infinity in the above series for a(n) given by Kotesovec gives the series expansion 1/e^2 = Sum_{k >= 0} (-1)^k*(k+3)*2^(k+3)/(k+5)!.
The sequence b(n) := (1/2)*n!*(n+2)*(n+5) satisfies the recurrence for a(n) given above by Pauly but with the starting values b(0) = 5 and b(1) = 9. This leads to the finite continued fraction expansion a(n) = (1/2)*n!*(n+2)*(n+5)( 1/(5 + 4/(1 + 2/(2 + 4/(3 + ... + 2*(n-1)/n)))) ), valid for n >= 2. Letting n -> infinity in the previous result gives the infinite continued fraction expansion 1/e^2 = 1/(5 + 4/(1 + 2/(2 + 4/(3 + ... + 2*(n-1)/(n + ...))))). Cf. A082031. (End)
a(n) = A087981(n+2)/(2*(n+1)). - Seiichi Manyama, Apr 25 2025

A209429 Numerator of l(n), where l(1)=1, l(2)=2, l(n)=l(n-1)+2*l(n-2)/n.

Original entry on oeis.org

1, 2, 8, 11, 71, 268, 2302, 2771, 29543, 172654, 2194624, 7533469, 111102841, 875654984, 1335478594, 16332117629, 307026528761, 3040884758542, 63303287929996, 345404844856129, 7886534621278669, 94005382576044068, 2335627560917144282, 7547413632563686237, 11923476834093824801

Offset: 1

N. J. A. Sloane, Mar 22 2012

			1, 2, 8/3, 11/3, 71/15, 268/45, 2302/315, 2771/315, 29543/2835, 172654/14175, 2194624/155925, 7533469/467775, 111102841/6081075, 875654984/42567525, ...

Szekeres, G. The average value of skew Hadamard matrices. Proceedings of the First Australian Conference on Combinatorial Mathematics (Univ. Newcastle, Newcastle, 1972), pp. 55--59. Univ. of Newcastle Res. Associates, Newcastle, 1972. MR0349708 (50 #2201)

G. C. Greubel, Table of n, a(n) for n = 1..500

Cf. A209430, A052127.

Numerator[RecurrenceTable[{a[1] == 1, a[2] == 2, a[n] == a[n - 1] + (2 a[n - 2])/n}, a, {n, 50}]] (* G. C. Greubel, Jan 02 2018 *)

A209430 Denominator of l(n), where l(1)=1, l(2)=2, l(n)=l(n-1)+2*l(n-2)/n.

Original entry on oeis.org

1, 1, 3, 3, 15, 45, 315, 315, 2835, 14175, 155925, 467775, 6081075, 42567525, 58046625, 638512875, 10854718875, 97692469875, 1856156927625, 9280784638125, 194896477400625, 2143861251406875, 49308808782358125, 147926426347074375, 217538862275109375, 4370553505709015625

Offset: 1

N. J. A. Sloane, Mar 22 2012

			1, 2, 8/3, 11/3, 71/15, 268/45, 2302/315, 2771/315, 29543/2835, 172654/14175, 2194624/155925, 7533469/467775, 111102841/6081075, 875654984/42567525, ...

Szekeres, G. The average value of skew Hadamard matrices. Proceedings of the First Australian Conference on Combinatorial Mathematics (Univ. Newcastle, Newcastle, 1972), pp. 55--59. Univ. of Newcastle Res. Associates, Newcastle, 1972. MR0349708 (50 #2201)

G. C. Greubel, Table of n, a(n) for n = 1..500

Cf. A209429, A052127.

Denominator[RecurrenceTable[{a[1]==1,a[2]==2,a[n]==a[n-1]+(2a[n-2])/n},a,{n,30}]] (* Harvey P. Dale, Mar 30 2014 *)

A357571 The sixth moment of an n X n random +-1 matrix.

Original entry on oeis.org

1, 1, 32, 1536, 282624, 66846720, 27053752320, 16104538275840, 13681567224299520, 15874223643851489280, 24412997036693834956800, 48514602066025722465484800, 121994703799547846503012761600, 381343447691461317926230740172800, 1459468400650603118890910517244723200

Offset: 0

Zelin Lv, Oct 03 2022

nonn

This sequence gives the expectation of the sixth moment of a random sign matrix (a matrix whose entries are independently set equal to -1 or 1 with equal probability) of size n X n.

Chai Wah Wu, Table of n, a(n) for n = 0..180
Dominik Beck, Zelin Lv and Aaron Potechin, The Sixth Moment of Random Determinants, arXiv:2206.11356 [math.CO], 2022-2023. See Table 1 p. 4.

Cf. A052127.

f6(n,m4,m6)=sum(j=0,n, binomial(n,j)*sum(a=0,j, binomial(j,a)*(m6-15)^a*(m4-3)^(j-a)*D(n,a,j-a)))
D(n,a,b)=prod(j=0,a+b-1,n-j)*sum(i=0,b, binomial(b,i)*C(i)*H(n,b-i,a,b))*P(n-a-b)
P(n)=n!*(n+2)!*(n+4)!/48
C(n)=if(n<2, n==0, (n-1)*(C(n-1)+15*C(n-2)))
H(n,j,a,b)=sum(x=1,j,binomial(j-1,x-1)*j!/x!*prod(y=0,x-1, 3*(n-a-b)-y))
\\ Charles R Greathouse IV, Oct 03 2022

a(n)={(n!)^2 * sum(j=0, n, sum(i=0, j, ((1+i)*(2+i)*(4+i)! / (48*(n-j)!)) * binomial(14+j+2*i,j-i) * (16)^(n-j) * (-2)^(j-i) ))} \\ Andrew Howroyd, Mar 16 2023

from fractions import Fraction
from math import factorial, comb
def A357571(n): return int(factorial(n)**2*sum(Fraction(1<<(n-j<<2),3*factorial(n-j))*sum((1+i)*(2+i)*factorial(4+i)*comb(14+j+(i<<1),j-i)*(-1 if (j-i)&1 else 1)<<(j-i) for i in range(j+1)) for j in range(n+1)))>>4 # Chai Wah Wu, Apr 20 2023

a(n) = (n!)^2 * Sum_{j=0..n} Sum_{i=0..j} ((1+i)*(2+i)*(4+i)! / (48*(n-j)!)) * binomial(14+j+2*i,j-i) * (m_6-15*m_4+30)^(n-j) * (m_4-3)^(j-i) where m_6 = m_4 = 1.

G.f.: A(t) = (e^(t*(m_6 - 15*m_4 + 30)) / (48*(1 + 3*t - m_4*t)^15)) * Sum_{i>=0} ((1+i)*(2+i)*(4+i)! * t^i / (1 + 3*t - m_4*t)^(3*i)) where m_6 = m_4 = 1.

a(0)=1 prepended and some terms corrected by Alois P. Heinz, Apr 19 2023

A362413 The second moment of an n X n symmetric random +-1 matrix.

Original entry on oeis.org

1, 1, 2, 8, 44, 244, 1744, 13768, 127952, 1270736, 14384096, 172799296, 2306400832, 32442943168, 498547591424, 8031916728704, 139611091407104, 2533449773986048, 49133884886866432, 991341134236389376, 21218511171980205056, 471083434031674336256

Offset: 0

Zelin Lv, Apr 18 2023

nonn

This sequence gives the expectation of the second moment of a random symmetric sign matrix of size n X n.

Zelin Lv, On The Moments of Random Determinants, Master Thesis, the University of Chicago.
I. G. Zhurbenko, Moments of random determinants, Theory of Probability & Its Application, 13(4):682-686, 1968.

Alois P. Heinz, Table of n, a(n) for n = 0..448

Cf. A052127, A357571.

a:= n-> `if`(n=0, 1, q(n)*(n-1)!):
p:= n-> `if`(n<3, 1, 3-irem(n, 2)):
q:= proc(n) option remember;
      p(n)+add(p(n-i)*q(i)/i, i=1..n-1)
    end:
seq(a(n), n=0..21);  # Alois P. Heinz, Apr 19 2023

a[n_] := If[n == 0, 1, q[n]*(n-1)!];
p[n_] := If[n < 3, 1, 3-Mod[n, 2]];
q[n_] := q[n] = p[n] + Sum[p[n-i]*q[i]/i, {i, 1, n-1}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 24 2025, after Alois P. Heinz *)

from math import factorial
from fractions import Fraction
from functools import lru_cache
@lru_cache(maxsize=None)
def A362413(n): return int(((1 if n<=2 else (2 if n&1 else 3))+sum(Fraction((1 if n-i<=2 else (2 if n-i&1 else 3))*A362413(i),factorial(i)) for i in range(1,n)))*factorial(n-1)) if n else 1 # Chai Wah Wu, Apr 20 2023

x = LazyPowerSeriesRing(QQ, "x").gen()
egf = exp(-x * (x + 1)) / sqrt((x + 1) * (1 - x)^5)
[egf[n] * factorial(n) for n in range(22)]  # Peter Luschny, Apr 20 2023

f^(sym)_2(n) = q(n) * (n-1)!, where
p(n) =
1, if n <= 2
2, if n >= 3 and n is odd
3, if n >= 4 and n is even
q(n) = p(n) + Sum_{i=1..n-1}(p(i) * q(n-i)) / (n-i).
E.g.f.: exp(-x*(x+1))/sqrt((x+1)*(1-x)^5). - Alois P. Heinz, Apr 19 2023
a(n) ~ 4*n^(n+2)/ (3*exp(n+2)). - Vaclav Kotesovec, Apr 20 2023
a(n) = (p(n) + Sum_{i=1..n-1} p(n-i) * a(i)/i! ) * (n-1)!. - Chai Wah Wu, Apr 20 2023

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A052124 Expansion of e.g.f. exp(-2*x)/(1-x)^3.

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A209429 Numerator of l(n), where l(1)=1, l(2)=2, l(n)=l(n-1)+2*l(n-2)/n.

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A209430 Denominator of l(n), where l(1)=1, l(2)=2, l(n)=l(n-1)+2*l(n-2)/n.

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A357571 The sixth moment of an n X n random +-1 matrix.

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A362413 The second moment of an n X n symmetric random +-1 matrix.

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