login
Expansion of the series reversion of x*(1 + 4*x + x^2)/(1 - x)^4.
0

%I #8 Jan 25 2023 13:10:36

%S 1,-8,101,-1544,26190,-474144,8975229,-175492664,3516970490,

%T -71858843264,1491301438354,-31349284476496,666133734882748,

%U -14284509655611840,308734263333717021,-6718525508918998872,147085140049822666626,-3237191565662618280384,71584853778205231503750

%N Expansion of the series reversion of x*(1 + 4*x + x^2)/(1 - x)^4.

%C Reversion of g.f. for the cubes (A000578).

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SeriesReversion.html">Series Reversion</a>

%H <a href="/index/Res#revert">Index entries for reversions of series</a>

%F G.f. A(x) satisfies: A(x)*(1 + 4*A(x) + A(x)^2)/(1 - A(x))^4 = x.

%F a(n) ~ -(-1)^n * (5*sqrt(6) - 12) * 2^(3*n-2) * 3^n / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Nov 11 2017

%F D-finite with recurrence 7*n*(n-1)*(n+1)*a(n) +4*n*(n-1)*(58*n-83)*a(n-1) -36*(n-1)*(16*n^2-80*n+155)*a(n-2) +432*(-96*n^3+720*n^2-1794*n+1495)*a(n-3) +6912*(4*n-15)*(2*n-7)*(4*n-13)*a(n-4)=0. - _R. J. Mathar_, Jan 25 2023

%t Rest[CoefficientList[InverseSeries[Series[x (1 + 4 x + x^2)/(1 - x)^4, {x, 0, 19}], x], x]]

%Y Cf. A000578, A007297, A263843.

%K sign

%O 1,2

%A _Ilya Gutkovskiy_, Aug 25 2017