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%I #24 Feb 16 2024 09:53:45
%S 1,2,8,38,201,1134,6688,40734,254237,1617572,10452416,68408626,
%T 452530659,3020870352,20324167488,137672551630,938154745773,
%U 6426806842566,44234352581896,305743015718028,2121318029754770,14769052147618740,103148538125870880
%N Expansion of (1/x) * Series_Reversion( x*(1+x)*(1-x)^3 ).
%H <a href="/index/Res#revert">Index entries for reversions of series</a>
%F a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(n+k,k) * binomial(4*n-k+2,n-k).
%F a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+k,k) * binomial(3*n-2*k+1,n-2*k). - _Seiichi Manyama_, Jan 18 2024
%F a(n) = (1/(n+1)) * [x^n] 1/( (1+x) * (1-x)^3 )^(n+1). - _Seiichi Manyama_, Feb 16 2024
%o (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n+k, k)*binomial(4*n-k+2, n-k))/(n+1);
%o (SageMath)
%o def A365751(n):
%o h = binomial(4*n + 2, n) * hypergeometric([-n, n + 1], [-4 * n - 2], -1) / (n + 1)
%o return simplify(h)
%o print([A365751(n) for n in range(23)]) # _Peter Luschny_, Sep 20 2023
%Y Cf. A063020, A365752, A365753.
%Y Cf. A352373.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Sep 18 2023
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