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

%I #13 Apr 30 2024 06:10:12

%S 1,0,0,2,2,2,15,32,53,192,527,1152,3327,9578,24217,66528,190357,

%T 515692,1421172,4036034,11272501,31489762,89370575,253106188,

%U 715642419,2038291672,5816775442,16592350656,47490009821,136246784272,391111252072,1124779108330

%N Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1-x+x^3)^2 ).

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

%F a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(2*n+2,k) * binomial(n-2*k-1,n-3*k).

%o (PARI) my(N=40, x='x+O('x^N)); Vec(serreverse(x*(1-x)^2/(1-x+x^3)^2)/x)

%o (PARI) a(n, s=3, t=2, u=2) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u-t+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);

%Y Cf. A369231, A372416.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Apr 29 2024