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

%I #11 Jan 17 2024 09:35:33

%S 1,0,0,3,3,3,33,72,120,583,1731,3888,13759,44775,119793,381220,

%T 1250328,3682284,11455153,37174428,114947724,359381467,1157319135,

%U 3663615552,11581104121,37220909916,119192219799,380580143110,1225279436706,3948906772872,12705801908002

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

%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(3*n+3,k) * binomial(n-2*k-1,n-3*k).

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

%o (PARI) a(n, s=3, t=3, u=3) = 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. A366052, A369231.

%Y Cf. A369081.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Jan 17 2024