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A365880 Expansion of (1/x) * Series_Reversion( x*(1+x)^4*(1-x)^5 ).

%I A365880 #18 Feb 16 2024 09:52:00
%S A365880 1,1,6,21,116,566,3176,17501,101391,590756,3519782,21163038,128845344,
%T A365880 790810400,4894134376,30486741869,191068074202,1203710067455,
%U A365880 7619193325238,48430121151156,309011352878208,1978450305442086,12706836843595840
%N A365880 Expansion of (1/x) * Series_Reversion( x*(1+x)^4*(1-x)^5 ).
%H A365880 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A365880 a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(4*n+k+3,k) * binomial(6*n-k+4,n-k).
%F A365880 a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(4*n+k+3,k) * binomial(2*n-2*k,n-2*k). - _Seiichi Manyama_, Jan 18 2024
%F A365880 a(n) = (1/(n+1)) * [x^n] 1/( (1+x)^4 * (1-x)^5 )^(n+1). - _Seiichi Manyama_, Feb 16 2024
%o A365880 (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(4*n+k+3, k)*binomial(6*n-k+4, n-k))/(n+1);
%o A365880 (SageMath)
%o A365880 def A365880(n):
%o A365880     h = binomial(6*n + 4, n) * hypergeometric([-n, 4*(n + 1)], [-6 * n - 4], -1) / (n + 1)
%o A365880     return simplify(h)
%o A365880 print([A365880(n) for n in range(23)])  # _Peter Luschny_, Sep 21 2023
%Y A365880 Cf. A365753, A365856, A365879.
%K A365880 nonn
%O A365880 0,3
%A A365880 _Seiichi Manyama_, Sep 21 2023