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

%I #16 Jan 13 2024 10:45:21

%S 1,3,15,94,660,4959,38995,316875,2639754,22423292,193484208,

%T 1691190228,14942632450,133242614565,1197520200870,10836727044255,

%U 98656011543816,902936341411170,8303218554134769,76679352910367832,710839322080978272,6612557820697157410

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

%H Seiichi Manyama, <a href="/A368964/b368964.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)} 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, binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);

%Y Cf. A368963, A368972.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 10 2024