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Expansion of (1/x) * Series_Reversion( x / (1+x)^2 * (1-x^3)^2 ).
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%I #13 Feb 16 2024 09:54:01

%S 1,2,5,16,62,264,1172,5342,24905,118410,572167,2801354,13865237,

%T 69258500,348698784,1767724720,9015710574,46227736956,238159867070,

%U 1232206495528,6399778252336,33354634754364,174390047681360,914414985920664,4807481173042396

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

%F a(n) = (1/(n+1)) * [x^n] ( (1+x)^2 / (1-x^3)^2 )^(n+1). - _Seiichi Manyama_, Feb 16 2024

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

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

%Y Cf. A396298.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 22 2024