cp's OEIS Frontend

A369694 Expansion of (1/x) * Series_Reversion( x * ((1-x)^3-x^2) ).

%I A369694 #13 Jan 29 2024 09:00:15
%S A369694 1,3,16,106,786,6244,51964,447201,3947306,35538668,325098696,
%T A369694 3013060258,28232408848,267003169668,2545341982728,24433290332007,
%U A369694 235967943943224,2291147902820524,22352525061549604,219006814853751540,2154083325737401740
%N A369694 Expansion of (1/x) * Series_Reversion( x * ((1-x)^3-x^2) ).
%H A369694 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A369694 a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+k,k) * binomial(4*n+k+2,n-2*k).
%F A369694 a(n) ~ sqrt((60 + (220324 - 42734*sqrt(2))^(1/3) + (220324 + 42734*sqrt(2))^(1/3)) / (138*Pi)) * (((4/23)*(22 + 3*(293 - 92*sqrt(2))^(1/3) + 3*(293 + 92*sqrt(2))^(1/3)))^n / n^(3/2)). - _Vaclav Kotesovec_, Jan 29 2024
%t A369694 CoefficientList[InverseSeries[Series[x*((1-x)^3 - x^2), {x, 0, 30}], x]/x, x](* _Vaclav Kotesovec_, Jan 29 2024 *)
%o A369694 (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^3-x^2))/x)
%o A369694 (PARI) a(n) = sum(k=0, n\2, binomial(n+k, k)*binomial(4*n+k+2, n-2*k))/(n+1);
%Y A369694 Cf. A001002, A151374.
%Y A369694 Cf. A369114, A369161, A369215.
%K A369694 nonn
%O A369694 0,2
%A A369694 _Seiichi Manyama_, Jan 29 2024