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

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

Offset: 0

Author

Fung Lam, Jan 16 2014

Keywords

• nonn
• easy

Comments

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

Links

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

Crossrefs

Cf. A107841, A235349.

Programs

PARI

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

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 = 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)

Formula

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

Extensions

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