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Expansion of (1/x) * Series_Reversion( x / ((1+x) * (1+x+x^3)) ).
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%I #11 Jan 24 2024 05:55:31

%S 1,2,5,15,51,187,718,2844,11530,47612,199576,847013,3632468,15717041,

%T 68527255,300780438,1327939406,5893299392,26275243626,117635107818,

%U 528631769323,2383660351991,10781500113896,48903885040638,222400899237943,1013841791472632

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

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

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

%Y Cf. A071879, A369482.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 23 2024