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A372039
Expansion of ( 1 + 9*x*(1 + x) )^(1/3).
0
1, 3, -6, 27, -144, 837, -5139, 32778, -215001, 1440747, -9818820, 67834665, -473945580, 3342743235, -23766448545, 170148578130, -1225477405485, 8873126329095, -64547392633740, 471509782020405, -3457212506428230, 25434642838306185, -187694935991201745
OFFSET
0,2
FORMULA
a(n) = Sum_{k=0..n} 9^k * binomial(1/3,k) * binomial(k,n-k).
a(n) ~ (-1)^(n+1) * Gamma(1/3) * 5^(1/6) * 3^(n - 1/2) * phi^(2*n - 2/3) / (2*Pi*n^(4/3)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Apr 19 2024
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec((1+9*x*(1+x))^(1/3))
(PARI) a(n) = sum(k=0, n, 9^k*binomial(1/3, k)*binomial(k, n-k));
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Apr 16 2024
STATUS
approved