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 A235349 Series reversion of x*(1-x-2*x^2)/(1-x). 3
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified September 17 15:10 EDT 2024. Contains 375987 sequences. (Running on oeis4.)