A365879 - OEIS

%I #22 Feb 16 2024 09:52:04

%S 1,2,10,54,332,2162,14734,103630,746857,5486206,40926152,309202686,

%T 2361065920,18192978966,141280871840,1104603758526,8687878404289,

%U 68692224882620,545681467048132,4353096328518810,34858239962177764,280095777427596008

%N Expansion of (1/x) * Series_Reversion( x*(1+x)^3*(1-x)^5 ).

%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(3*n+k+2,k) * binomial(6*n-k+4,n-k).

%F a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(3*n+k+2,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)^3 * (1-x)^5 )^(n+1). - _Seiichi Manyama_, Feb 16 2024

%o (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(3*n+k+2, k)*binomial(6*n-k+4, n-k))/(n+1);

%o (SageMath)

%o def A365879(n):

%o     h = binomial(6*n + 4, n) * hypergeometric([-n, 3*(n + 1)], [-6 * n - 4], -1) / (n + 1)

%o     return simplify(h)

%o print([A365879(n) for n in range(22)])  # _Peter Luschny_, Sep 21 2023

%Y Cf. A365753, A365856, A365880.

%Y Cf. A365878, A368079.

%Y Cf. A370270.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Sep 21 2023