A365751 Expansion of (1/x) * Series_Reversion( x*(1+x)*(1-x)^3 ).
1, 2, 8, 38, 201, 1134, 6688, 40734, 254237, 1617572, 10452416, 68408626, 452530659, 3020870352, 20324167488, 137672551630, 938154745773, 6426806842566, 44234352581896, 305743015718028, 2121318029754770, 14769052147618740, 103148538125870880
Offset: 0
Keywords
Programs
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PARI
a(n) = sum(k=0, n, (-1)^k*binomial(n+k, k)*binomial(4*n-k+2, n-k))/(n+1);
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SageMath
def A365751(n): h = binomial(4*n + 2, n) * hypergeometric([-n, n + 1], [-4 * n - 2], -1) / (n + 1) return simplify(h) print([A365751(n) for n in range(23)]) # Peter Luschny, Sep 20 2023
Formula
a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(n+k,k) * binomial(4*n-k+2,n-k).
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
a(n) = (1/(n+1)) * [x^n] 1/( (1+x) * (1-x)^3 )^(n+1). - Seiichi Manyama, Feb 16 2024