A247029 - OEIS

%I #8 Nov 18 2017 05:25:27

%S 1,3,-18,180,-2187,29484,-424116,6377292,-99034650,1576075644,

%T -25569752274,421325812440,-7031733125508,118620405322020,

%U -2019349799669160,34647126360607440,-598525520999144643,10401492640172342940,-181721630178565389900,3189811189331825319492

%N G.f. satisfies: A(x) = A(x)^4 - 9*x.

%H Vaclav Kotesovec, <a href="/A247029/b247029.txt">Table of n, a(n) for n = 0..500</a>

%F G.f.: x / Series_Reversion( x*(1 + 9*x)^(1/3) ).

%F Recurrence: (n-2)*(n-1)*n*a(n) = -216*(2*n - 5)*(4*n - 13)*(4*n - 7)*a(n-3). - _Vaclav Kotesovec_, Nov 18 2017

%F a(n) ~ -(-1)^n * 2^(8*n/3 - 13/6) * 3^n / (sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Nov 18 2017

%e G.f.: A(x) = 1 + 3*x - 18*x^2 + 180*x^3 - 2187*x^4 + 29484*x^5 - 424116*x^6 +...

%e where

%e A(x)^4 = 1 + 12*x - 18*x^2 + 180*x^3 - 2187*x^4 + 29484*x^5 - 424116*x^6 +...

%t FullSimplify[Table[-(-1)^n * 3^(2*n-1) * 4^(n-1) * Gamma[n/3 + 1/6] * Gamma[2*n/3 - 1/6] / (Pi * Gamma[n + 1]), {n, 0, 20}]] (* _Vaclav Kotesovec_, Nov 18 2017 *)

%o (PARI) {a(n)=polcoeff(x/serreverse(x*(1+9*x +x^2*O(x^n))^(1/3)), n)}

%o for(n=0, 25, print1(a(n), ", "))

%Y Cf. A224884.

%K sign

%O 0,2

%A _Paul D. Hanna_, Sep 09 2014