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G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^3.
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%I #14 Mar 27 2024 08:54:08

%S 1,3,15,82,477,2901,18235,117555,773085,5166478,34987170,239570655,

%T 1655933060,11538839130,80971109712,571702698185,4058556404958,

%U 28951715755830,207424064434502,1491898838023884,10768487956456506,77977009814421534,566310026687320290

%N G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^3.

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

%F G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A307678.

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

%Y Cf. A045868, A371517, A371520, A371521.

%Y Cf. A270386, A307678.

%Y Cf. A371483.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 26 2024