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Expansion of 1/(1 - 9*x/(1-x))^(2/3).
3

%I #11 Oct 21 2024 05:06:43

%S 1,6,51,456,4191,39174,370329,3529284,33838854,325978044,3152058630,

%T 30572797920,297294956070,2897207397420,28286321963370,

%U 276611636831640,2708781551458665,26559205696513590,260695647288540915,2561413004129212440,25188928968792165495

%N Expansion of 1/(1 - 9*x/(1-x))^(2/3).

%F a(0) = 1; a(n) = 3 * Sum_{k=0..n-1} (2+k/n) * a(k).

%F a(n) = ((11*n-5)*a(n-1) - 10*(n-2)*a(n-2))/n for n > 1.

%F a(n) = Sum_{k=0..n} (-9)^k * binomial(-2/3,k) * binomial(n-1,n-k).

%F a(n) ~ Gamma(1/3) * 3^(11/6) * 2^(n - 5/3) * 5^(n - 2/3) / (Pi * n^(1/3)). - _Vaclav Kotesovec_, Oct 21 2024

%o (PARI) a(n) = sum(k=0, n, (-9)^k*binomial(-2/3, k)*binomial(n-1, n-k));

%Y Cf. A052268, A361375, A377234, A377235.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Oct 21 2024