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

%I #16 Jan 13 2024 10:45:33

%S 1,3,15,88,564,3819,26851,194025,1431498,10735548,81580008,626697786,

%T 4858272450,37954323885,298487957670,2361025981335,18770449480056,

%U 149897172319290,1201831832357041,9670416882346848,78062823843714528,631988009034161246

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

%C a(702) is negative.

%H Seiichi Manyama, <a href="/A368972/b368972.txt">Table of n, a(n) for n = 0..1000</a>

%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)} (-1)^k * binomial(3*n+k+2,k) * binomial(4*n-2*k+2,n-3*k).

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x+x^3)^3)/x)

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

%Y Cf. A368964.

%K sign

%O 0,2

%A _Seiichi Manyama_, Jan 10 2024