cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A235347 Series reversion of x*(1-3*x^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

Views

Author

Fung Lam, Jan 10 2014

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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

Formula

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)

A235349 Series reversion of x*(1-x-2*x^2)/(1-x).

Original entry on oeis.org

0, 1, 0, 2, 2, 14, 30, 146, 434, 1862, 6470, 26586, 99946, 406366, 1593774, 6492450, 26100578, 106979894, 436906902, 1803472874, 7446478746, 30945624910, 128821054846, 538584390834, 2256485249682, 9483898177574
Offset: 0

Views

Author

Fung Lam, Jan 16 2014

Keywords

Comments

Derived turbulence series from A235347.

Links

Crossrefs

Cf. A235347, A235348, A235350, A235351, A235352.

Programs

  • Mathematica
    CoefficientList[InverseSeries[Series[x*(1-x-2*x^2)/(1-x), {x, 0, 20}], x],x] (* Vaclav Kotesovec, Jan 22 2014 *)
  • PARI
    Vec(serreverse(x*(1-x-2*x^2)/(1-x)+O(x^66))) \\ Joerg Arndt, Jan 17 2014
  • Python
    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 += a[i]*a[j]
    g = 0
    for i in range (1, m-1):
        for j in range (1, m-1):
            for k in range (1, m-1):
                if (i+j+k)%m == 0 and (i+j+k) <= m:
                    g += a[i]*a[j]*a[k]
    y = 2*g + d - a[m-1]
    a.append(y)
print(a)

Formula

G.f.: ( exp(4*Pi*i/3)*u + exp(2*Pi*i/3)*v - 1/6 )/x, where i=sqrt(-1),
u = 1/6*(-10-63*x+3*sqrt(-24*x^3+357*x^2+42*x-27))^(1/3), and
v = 1/6*(-10-63*x-3*sqrt(-24*x^3+357*x^2+42*x-27))^(1/3).
a(n) ~ sqrt((1-s)^3 / (2*s*(3 - 3*s + s^2))) / (2*sqrt(Pi) * n^(3/2) * r^(n-1/2)), where s = 0.31472177038151893868... is the root of the equation 1-2*s-5*s^2+4*s^3 = 0, and r = s*(1-s-2*s^2)/(1-s) = 0.22374229727550306625... - Vaclav Kotesovec, Jan 23 2014
D-finite with recurrence 117*n*(n-1)*a(n) -7*(n-1)*(35*n-66)*a(n-1) +21*(-69*n^2+269*n-254)*a(n-2) +(937*n^2-6403*n+10920)*a(n-3) -28*(n-4)*(2*n-9)*a(n-4)=0. - R. J. Mathar, Mar 24 2023

Extensions

Prepended a(0)=0 to adapt to offset 0, Joerg Arndt, Jan 23 2014
b-file shifted for offset 0, Vaclav Kotesovec, Jan 23 2014

A235350 Series reversion of x*(1-2*x-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

Views

Author

Fung Lam, Jan 16 2014

Keywords

Comments

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).

Links

Crossrefs

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

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x*(1-2*x-x^2)/(1-x^2), {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Jan 29 2014 *)
  • PARI
    Vec(serreverse(x*(1-2*x-x^2)/(1-x^2)+O(x^66))) \\ Joerg Arndt, Jan 17 2014
  • Python
    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

Formula

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
Showing 1-3 of 3 results.

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