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

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

0,4

Comments

Derived turbulence series from A235347.

Links

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

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

Mathematica

CoefficientList[InverseSeries[Series[x*(1-x-2*x^2)/(1-x), {x, 0, 20}], x], x] (* Vaclav Kotesovec, Jan 22 2014 *)

Prog

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

Crossrefs

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

Sequence in context: A187734 A151353 A151437 * A226157 A264508 A194689

Adjacent sequences: A235346 A235347 A235348 * A235350 A235351 A235352

Keyword

nonn,easy,changed

Author

Fung Lam, Jan 16 2014

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

Status

approved