Fung Lam - cp's OEIS frontend

A240880 Expansion of g.f.: (-1 + sqrt(1+12x+48x^2)) / (6*x).

1, 1, -6, 33, -162, 666, -1836, -2079, 79542, -741474, 4907628, -24837030, 82449900, 53319060, -3741922008, 38613958497, -274566158298, 1475669401398, -5211777090564, -2356585871778, 240686500011588, -2593621485808596, 19047621883804056, -105353643788834598

Offset: 0

Fung Lam, May 01 2014

This sequence is the member (q=-3) of a class of generalized Catalan numbers (see A000108), with g.f. (1-sqrt(1-q*4*x*(1-(q-1)*x)))/(2*q*x), q<>0.

Fung Lam, Table of n, a(n) for n = 0..1000

Cf. A000108, A258723.

CoefficientList[Series[(Sqrt[1+12x+48x^2]-1)/(6x),{x,0,30}],x]  (* Harvey P. Dale, May 24 2022 *)

G.f.: (-1 + sqrt(1+12*x+48*x^2)) / (6*x).
D-finite with recurrence: (n+3)*a(n+2)+6*(2*n+3)*a(n+1)+48*n*a(n)=0, a(0)=1, a(1)=1.
Lim sup n->infinity |a(n)|^(1/n) = 4*sqrt(3) = 6.9282... - Vaclav Kotesovec, May 02 2014
a(n) ~ 3^(n/2-1)*4^n / (n^(3/2)*sqrt(Pi)) * (sqrt(3)*cos(5*Pi*n/6) + 3*sin(5*Pi*n/6) - (15*sqrt(3)*cos(5*Pi*n/6) + 9*sin(5*Pi*n/6))/(8*n)). - Vaclav Kotesovec, May 02 2014

A240881 Chebyshev transform of A107841.

Original entry on oeis.org

1, 2, 9, 58, 401, 2952, 22759, 181358, 1481751, 12346102, 104505959, 896170608, 7768885801, 67972510202, 599449125609, 5323095489058, 47555513297801, 427127946025752, 3854618439044959, 34934658168463958, 317834095671077751, 2901725605879035502, 26575914921615695759

Offset: 0

Fung Lam, Apr 29 2014

This is the Chebyshev transform over the positive strip 0<=x<=1. A160852 may be viewed as the Chebyshev transform over the negative strip -1<=x<=0.

Fung Lam, Table of n, a(n) for n = 0..1000

Cf. A107841, A160852.

CoefficientList[Series[(1+x+x^2 - Sqrt[1-10*x+3*x^2-10*x^3+x^4])/(6*x*(1+x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 30 2014 *)

x='x+O('x^50); Vec((1+x+x^2 - sqrt(1-10*x+3*x^2-10*x^3+x^4))/(6*x*(1+x^2))) \\ G. C. Greubel, Apr 05 2017

G.f.: (1+x+x^2 - sqrt(1-10*x+3*x^2-10*x^3+x^4))/(6*x*(1+x^2)).
G.f.: F(x/(1+x^2)), where F(x) is the g.f. of A107841.
Recurrence: (n+1)*a(n) = (5-n)*a(n-6) + 5*(2*n-7)*a(n-5) + (11-4*n)*a(n-4)
    + 20*(n-2)*a(n-3) + (5-4*n)*a(n-2) + 5*(2*n-1)*a(n-1), n>=6.
a(n) ~ (sqrt(45+20*sqrt(6))/2+sqrt(6)+5/2)^n*sqrt(120-30*sqrt(6)+2*sqrt(30*(6196*sqrt(6)-15159)))/(12*sqrt(Pi*n^3)).

A240879 Self-convolution of Sum(binomial(2*n, i), i=0..n).

Original entry on oeis.org

1, 6, 31, 150, 699, 3178, 14198, 62604, 273235, 1182786, 5085666, 21743956, 92522206, 392066340, 1655432524, 6967724312, 29245179267, 122442487474, 511487386730, 2132341655556, 8873167793578, 36861311739308, 152895342950196, 633290273209000, 2619653638855214, 10823294835350388

Offset: 0

Fung Lam, Apr 13 2014

nonn

Fung Lam, Table of n, a(n) for n = 0..1000

Cf. A032443.

CoefficientList[Series[((1/Sqrt[1-4*x] + 1/(1-4*x))/2)^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 16 2014 *)

G.f. = (g.f. of A032443)^2.
n*a(n) = 32*(2*n-3)*a(n-3) + 48*(1-n)*a(n-2) + 6*(2*n-1)*a(n-1).
Asymptotics: a(n) ~ 2^(2*n)*((n+2)/4 + sqrt(n/Pi)).
Recurrence: (n-2)*n*a(n) = 2*n*(4*n-7)*a(n-1) - 8*(n-1)*(2*n-1)*a(n-2). - Vaclav Kotesovec, Apr 16 2014

A238755 Second convolution of A065096.

Original entry on oeis.org

0, 0, 1, 12, 98, 684, 4403, 27048, 161412, 945288, 5466549, 31340628, 178604998, 1013573652, 5735117479, 32385232272, 182622362504, 1028897389008, 5793703249449, 32615362319580, 183593293074730, 1033535639454780, 5819389057957211, 32775522041862072, 184658694508103180

Offset: 0

Fung Lam, Mar 04 2014

Fung Lam, Table of n, a(n) for n = 0..1300

Cf. A065096, A000108, A001003.

CoefficientList[Series[(1-3*x-Sqrt[1-6*x+x^2])^4/(16*x^3)^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 05 2014 *)

x='x+O('x^50); concat([0,0], Vec((1-3*x-sqrt(1-6*x+x^2))^4/(16*x^3)^2)) \\ G. C. Greubel, Apr 05 2017

G.f. = (G.f. of A065096)^2.
Recurrence: (n+6)*a(n) = 225*(6-n)*a(n-8) + 1020*(2*n-9)*a(n-7) + 5164*(3-n)*a(n-6) + 76*(78*n-117)*a(n-5) - 3590*n*a(n-4) + 36*(34*n+51)*a(n-3) - 236*(n+3)*a(n-2) + 12*(2*n+9)*a(n-1), n>=8.
Recurrence (of order 2): (n-2)*(n+6)*a(n) = 3*(n+1)*(2*n+3)*a(n-1) - n*(n+1)*a(n-2). - Vaclav Kotesovec, Mar 05 2014
a(n) ~ (3*sqrt(2)-4)^(7/2) * (3+2*sqrt(2))^(n+6) / (8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 05 2014

A238300 Fourth convolution of A107841.

Original entry on oeis.org

1, 8, 64, 520, 4304, 36232, 309504, 2677128, 23405520, 206522888, 1836913216, 16452907016, 148274884688, 1343569891720, 12233903203328, 111883174439304, 1027244073375312, 9465236716896264, 87498251217286720, 811252609543727624, 7542152541765899728, 70294794046928531848

Offset: 0

Fung Lam, Feb 25 2014

Fung Lam, Table of n, a(n) for n = 0..1000

Cf. A107841, A238299.

CoefficientList[Series[((1+x-Sqrt[1-10*x+x^2])/(6*x))^4,{x,0,20}],x] (* Vaclav Kotesovec, Feb 27 2014 *)

G.f.: (G.f. of A107841)^4.
Recurrence: (n+4)*a(n) = (8-n)*a(n-8) + 4*(4*n-26)*a(n-7) + 64*(5-n)*a(n-6) + 8*(2*n-7)*a(n-5) + 194*(n-2)*a(n-4) + 8*(2*n-1)*a(n-3) - 64*(n+1)*a(n-2) + 8*(2*n+5)*a(n-1), n>=8.
Recurrence (of order 2): n*(n+4)*(2*n+1)*a(n) = 20*n*(n+1)*(n+2)*a(n-1) - (n-2)*(n+2)*(2*n+3)*a(n-2). - Vaclav Kotesovec, Feb 27 2014
a(n) ~ 2*sqrt(35280+14403*sqrt(6)) * (5+2*sqrt(6))^n / (27 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 27 2014

A238299 Second convolution of A107841.

Original entry on oeis.org

1, 4, 24, 164, 1208, 9348, 74920, 616420, 5176296, 44182916, 382205048, 3343343268, 29523386968, 262826367748, 2356256046216, 21254326842596, 192766180154120, 1756758963727620, 16079466335134168, 147748236828875428, 1362397741935948024, 12603116216808465284, 116929440001191010664

Offset: 0

Fung Lam, Feb 25 2014

Fung Lam, Table of n, a(n) for n = 0..1000

Cf. A107841, A238300.

CoefficientList[Series[((1 + x - Sqrt[1 - 10*x + x^2])/(6*x))^2, {x, 0, 100}], x] (* Vaclav Kotesovec, Feb 27 2014 *)

G.f.: (G.f. of A107841)^2.
Recurrence: (n+2)*a(n) = (4-n)*a(n-4) + 4*(2*n-5)*a(n-3) + 18*(n-1)*a(n-2) + 4*(2*n+1)*a(n-1), n>=4.
Recurrence (of order 2): (n+2)*(2*n-1)*a(n) = 4*(5*n^2-2)*a(n-1) - (n-2)*(2*n+1)*a(n-2). - Vaclav Kotesovec, Feb 27 2014
a(n) ~ sqrt(360+147*sqrt(6)) * (5+2*sqrt(6))^n / (9 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 27 2014

A235350 Series reversion of x(1-2x-x^2)/(1-x^2).

Original entry on oeis.org

1, 2, 8, 42, 248, 1570, 10416, 71474, 503088, 3612226, 26353720, 194806458, 1455874792, 10982013250, 83504148192, 639360351074, 4925190101600, 38144591091970, 296837838901992, 2319880586624714, 18200693844341720, 143294043656426082, 1131747417739664528

Offset: 1

Fung Lam, Jan 16 2014

Derived series from A107841. The reversion has a quadratic power in x in the denominator. The general form reads x*(1-p*x-q*x^2)/(1-q*x^2).

Fung Lam, Table of n, a(n) for n = 1..1000

Cf. A107841, A235347, A235348, A235349, A235351, A235352.

Rest[CoefficientList[InverseSeries[Series[x*(1-2*x-x^2)/(1-x^2), {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Jan 29 2014 *)

Vec(serreverse(x*(1-2*x-x^2)/(1-x^2)+O(x^66))) \\ Joerg Arndt, Jan 17 2014

a = [0, 1]
for n in range(20):
    m = len(a)
    d = 0
    for i in range (1, m):
        for j in range (1, m):
            if (i+j)%(m-1) == 0 and (i+j) < m:
                d += a[i]*a[j]
    f = 0
    for i in range (1, m):
        for j in range (1, m):
            if (i+j)%m == 0 and (i+j) <= m:
                f += a[i]*a[j]
    g = 0
    for i in range (1, m):
        for j in range (1, m):
            for k in range (1, m):
                if (i+j+k)%m == 0 and (i+j+k) <= m:
                    g += a[i]*a[j]*a[k]
    y = g + 2*f - d
    a.append(y)
print(a[1:]) # Edited by Andrey Zabolotskiy, Sep 04 2024

G.f.: (exp(4*Pi*i/3)*u + exp(2*Pi*i/3)*v - 2/3)/x, where i=sqrt(-1),
u = 1/3*(-17+3*x-6*x^2+x^3+3*sqrt(-6+54*x-30*x^2+18*x^3-3*x^4))^(1/3), and
v = 1/3*(-17+3*x-6*x^2+x^3-3*sqrt(-6+54*x-30*x^2+18*x^3-3*x^4))^(1/3).
D-finite with recurrence 6*n*(n-1)*a(n) -(n-1)*(52*n-75)*a(n-1) +(2*n+3)*(5*n-11)*a(n-2) +2*(5*n^2-62*n+150)*a(n-3) +(-13*n^2+130*n-321)*a(n-4) +(7*n-37)*(n-6)*a(n-5) -(n-6)*(n-7)*a(n-6)=0. - R. J. Mathar, Mar 24 2023

A235348 Series reversion of x(1-2x-5*x^2)/(1-x^2).

Original entry on oeis.org

1, 2, 12, 82, 636, 5266, 45684, 409706, 3768132, 35346082, 336854844, 3252391170, 31746462732, 312755404818, 3105750620772, 31054695744570, 312404601250644, 3159598296022978, 32108181705850860, 327682918265502002, 3357089384702757276

Offset: 1

Fung Lam, Jan 13 2014

Sum of turbulence series A107841 and A235347.

Fung Lam, Table of n, a(n) for n = 1..1000

Rest[CoefficientList[InverseSeries[Series[x*(1-2*x-5*x^2)/(1-x^2), {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Jan 29 2014 *)

Vec( serreverse(x*(1-2*x-5*x^2)/(1-x^2) +O(x^66) ) ) \\ Joerg Arndt, Jan 14 2014

# R. J. Mathar, 2023-03-28
class A235348() :
    def _init_(self) :
        self.a = [1, 2, 12, 82, 636, 5266]
    def at(self, n):
        if n <= len(self.a):
            return self.a[n-1]
        else:
            rhs = -3*(n-1)*(160*n-237)*self.at(n-1) \
            +3*(-422*n**2+1721*n-1713)*self.at(n-2) \
            +2*(-67*n**2+388*n-552)*self.at(n-3) \
            +(137*n**2-1352*n+3279)*self.at(n-4) \
            +(7*n-37)*(n-6)*self.at(n-5) -(n-6)*(n-7)*self.at(n-6)
            rhs //= (-54*n*(n-1))
            self.a.append(rhs)
            return self.a[-1]
a235348 = A235348()
for n in range(1,12):
    print(a235348.at(n))
# a235348.

D-finite with recurrence 54*n*(n-1)*a(n) -3*(n-1)*(160*n-237)*a(n-1) +3*(-422*n^2+1721*n-1713)*a(n-2) +2*(-67*n^2+388*n-552)*a(n-3) +(137*n^2-1352*n+3279)*a(n-4) +(7*n-37)*(n-6)*a(n-5) -(n-6)*(n-7)*a(n-6)=0. - R. J. Mathar, Mar 24 2023

A235351 Series reversion of x(1-3x-2*x^2)/(1-x).

Original entry on oeis.org

0, 1, 2, 12, 84, 660, 5548, 48836, 444412, 4147220, 39471436, 381671204, 3738957148, 37028943860, 370123733932, 3729092573060, 37831802166076, 386135110256852, 3962278590508812, 40852572573083364, 423006921400424988, 4396894566694687924

Offset: 0

Fung Lam, Jan 16 2014

Derived turbulence series: combined series reversion of A107841 and A235349.

Fung Lam, Table of n, a(n) for n = 0..950

Cf. A107841, A235349.

my(x='x+O('x^25)); concat([0],Vec(serreverse(x*(1-3*x-2*x^2)/(1-x)))) \\ Joerg Arndt, Sep 01 2024

a = [0, 1]
for n in range(20):
    m = len(a)
    d = 0
    for i in range (1, m):
        for j in range (1, m):
            if (i+j)%m ==0 and (i+j) <= m:
                d = d + a[i]*a[j]
    g = 0
    for i in range (1, m):
        for j in range (1, m):
            for k in range (1, m):
                if (i+j+k)%m ==0 and (i+j+k) <= m:
                    g = g + a[i]*a[j]*a[k]
    y = 2*g + 3*d - a[m-1]
    a.append(y)
print(a)

G.f.: (exp(4*Pi*i/3)*u + exp(2*Pi*i/3)*v - 1/2)/x, where i=sqrt(-1),
u = 1/6*(-54-81*x+3*sqrt(-51+522*x+549*x^2-24*x^3))^(1/3), and
v = 1/6*(-54-81*x-3*sqrt(-51+522*x+549*x^2-24*x^3))^(1/3).
D-finite with recurrence 17*n*(n+1)*(11*n-17)*a(n) -n*(1914*n^2-3915*n+1513)*a(n-1) +(-2013*n^3+7137*n^2-7924*n+2640)*a(n-2) +4*(2*n-5)*(11*n-6)*(n-2)*a(n-3)=0. - R. J. Mathar, Jun 14 2016

a(0) = 0 prepended by Andrey Zabolotskiy, Aug 31 2024

A235347 Series reversion of x(1-3x^2)/(1-x^2) in odd-order powers.

Original entry on oeis.org

1, 2, 14, 130, 1382, 15906, 192894, 2427522, 31405430, 415086658, 5580629870, 76080887042, 1049295082630, 14613980359010, 205246677882078, 2903566870820610, 41337029956899222, 591796707042765954, 8514525059135909070, 123048063153362454402

Offset: 0

Fung Lam, Jan 10 2014

This sequence is implied in the solutions of magnetohydrodynamics equations in R^3 for incompressible, electrically-conducting fluids in the presence of a strong Lorentz force. a(n) = numbers of allowable magneto-vortical eddies in terms of initial conditions.

Georg Fischer, Table of n, a(n) for n = 0..1000 (first 940 terms from _Fung Lam_)
Fung Lam, On the Well-posedness of Magnetohydrodynamics Equations for Incompressible Electrically-Conducting Fluids, arXiv:1401.2029 [physics.flu-dyn], 2014-2023.

Cf. A027307, A107841, A235352 (same except for signs).

Order := 60 ;
solve(series(x*(1-3*x^2)/(1-x^2),x)=y,x) ;
convert(%,polynom) ;
seq(coeff(%,y,2*i+1),i=0..Order/2) ; # R. J. Mathar, Jul 20 2023

Table[(CoefficientList[InverseSeries[Series[x*(1-3*x^2)/(1-x^2),{x,0,40}],x],x])[[n]],{n,2,40,2}] (* Vaclav Kotesovec, Jan 29 2014 *)

v=Vec( serreverse(x*(1-3*x^2)/(1-x^2) +O(x^66) ) ); vector(#v\2,j,v[2*j-1]) \\ Joerg Arndt, Jan 14 2014

G.f.: (exp(4*Pi*i/3)*u + exp(2*Pi*i/3)*v + x/9)/x, where i=sqrt(-1),
u = (1/9)*(x^3 - 108 *x + 9*sqrt(-9 + 141*x^2 - 3*x^4))^(1/3), and
v = (1/9)*(x^3 - 108 *x - 9*sqrt(-9 + 141*x^2 - 3*x^4))^(1/3).
a(n) = [x^n] 2*Sum_{j = 1..n} ((Sum_{k = 1..n} a(k)*x^(2*k-1))^(2*j+1)), a(1) = 1, with offset by 1.
D-finite with recurrence 12*n*(2*n+1)*a(n) +(-382*n^2+391*n-90)*a(n-1) +3*(34*n^2-132*n+125)*a(n-2) -(2*n-5)*(n-3)*a(n-3)=0. - R. J. Mathar, Mar 24 2023
From Seiichi Manyama, Aug 09 2023: (Start)
a(n) = (-1)^n * Sum_{k=0..n} (-3)^k * binomial(n,k) * binomial(2*n+k+1,n) / (2*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} 2^(n-k) * binomial(n,k) * binomial(3*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 2^k * 3^(n-k) * binomial(n,k) * binomial(2*n,k-1) for n > 0. (End)
From Peter Bala, Sep 08 2024: (Start)
a(n) = 2*Jacobi_P(n-1, 1, n+1, 5)/n for n >= 1.
Second-order recurrence: 3*n*(2*n + 1)*(13*n - 17)*a(n) = (1222*n^3 - 2820*n^2 + 1877*n - 360)*a(n-1) - (n - 2)*(13*n - 4)*(2*n - 3)*a(n-2) with a(0) = 1 and a(1) = 2. (End)

cp's OEIS Frontend

User: Fung Lam

A240880 Expansion of g.f.: (-1 + sqrt(1+12*x+48*x^2)) / (6*x).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A240881 Chebyshev transform of A107841.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A240879 Self-convolution of Sum(binomial(2*n, i), i=0..n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A238755 Second convolution of A065096.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A238300 Fourth convolution of A107841.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A238299 Second convolution of A107841.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A235350 Series reversion of x*(1-2*x-x^2)/(1-x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A235348 Series reversion of x*(1-2*x-5*x^2)/(1-x^2).

Views

Author

Keywords

Comments

Links

Programs

Mathematica

A240880 Expansion of g.f.: (-1 + sqrt(1+12x+48x^2)) / (6*x).

A235350 Series reversion of x(1-2x-x^2)/(1-x^2).

A235348 Series reversion of x(1-2x-5*x^2)/(1-x^2).

A235351 Series reversion of x(1-3x-2*x^2)/(1-x).

A235347 Series reversion of x(1-3x^2)/(1-x^2) in odd-order powers.