A365856 Expansion of (1/x) * Series_Reversion( x*(1+x)^2*(1-x)^5 ).
1, 3, 17, 115, 863, 6903, 57687, 497683, 4398980, 39630305, 362562226, 3359252039, 31457036708, 297247495745, 2830725974514, 27140465365203, 261768686779800, 2538061348959000, 24724191398850125, 241862002342417585, 2374978445599884762
Offset: 0
Keywords
Programs
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PARI
a(n) = sum(k=0, n, (-1)^k*binomial(2*n+k+1, k)*binomial(6*n-k+4, n-k))/(n+1);
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SageMath
def A365856(n): h = binomial(6*n + 4, n) * hypergeometric([-n, 2*n + 2], [-6 * n - 4], -1) / (n + 1) return simplify(h) print([A365856(n) for n in range(21)]) # Peter Luschny, Sep 20 2023
Formula
a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(2*n+k+1,k) * binomial(6*n-k+4,n-k).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(2*n+k+1,k) * binomial(4*n-2*k+2,n-2*k). - Seiichi Manyama, Jan 18 2024
a(n) = (1/(n+1)) * [x^n] 1/( (1+x)^2 * (1-x)^5 )^(n+1). - Seiichi Manyama, Feb 16 2024