A365879 Expansion of (1/x) * Series_Reversion( x*(1+x)^3*(1-x)^5 ).
1, 2, 10, 54, 332, 2162, 14734, 103630, 746857, 5486206, 40926152, 309202686, 2361065920, 18192978966, 141280871840, 1104603758526, 8687878404289, 68692224882620, 545681467048132, 4353096328518810, 34858239962177764, 280095777427596008
Offset: 0
Keywords
Programs
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PARI
a(n) = sum(k=0, n, (-1)^k*binomial(3*n+k+2, k)*binomial(6*n-k+4, n-k))/(n+1);
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SageMath
def A365879(n): h = binomial(6*n + 4, n) * hypergeometric([-n, 3*(n + 1)], [-6 * n - 4], -1) / (n + 1) return simplify(h) print([A365879(n) for n in range(22)]) # Peter Luschny, Sep 21 2023
Formula
a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(3*n+k+2,k) * binomial(6*n-k+4,n-k).
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
a(n) = (1/(n+1)) * [x^n] 1/( (1+x)^3 * (1-x)^5 )^(n+1). - Seiichi Manyama, Feb 16 2024