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Expansion of (1/x) * Series_Reversion( x * (1-x)^3 / (1+x^3)^3 ).
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%I #17 Feb 15 2024 04:17:40

%S 1,3,15,94,657,4902,38233,307953,2541831,21386810,182754162,

%T 1581699162,13836248406,122139271098,1086638457429,9733419373534,

%U 87707244737511,794505072627735,7231017033165776,66089527981542462,606340568510978940,5582088822346925210

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

%H P. Bala, <a href="/A251592/a251592.pdf">Fractional iteration of a series inversion operator</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)} binomial(3*n+3,k) * binomial(4*n-3*k+2,n-3*k).

%F a(n) = (1/(n+1)) * [x^n] ( 1/(1-x)^3 * (1+x^3)^3 )^(n+1). - _Seiichi Manyama_, Feb 14 2024

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^3/(1+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+1)*(n+1)-s*k-2, n-s*k))/(n+1);

%Y Cf. A369268, A369269.

%Y Cf. A365843, A369264.

%Y Cf. A369232.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 18 2024