A371987 - OEIS

%I #10 Apr 15 2024 09:45:03

%S 1,6,-18,90,-486,2430,-8586,-17982,841266,-12165066,136875582,

%T -1348875990,12016318410,-96794708562,685263211974,-3870181566702,

%U 10180063779426,147487856352102,-3442575733736562,47851939835741178,-546779680526987910,5515345957243519710

%N G.f. A(x) satisfies A(x) = ( 1 + 9*x*(1 + A(x)) )^(1/3).

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

%F G.f.: (6*2^(1/3)*x + 2^(2/3) * (1 + 9*x + sqrt(1+9*x*(2+3*(3-4*x)*x)))^(2/3)) / (2*(1 + 9*x + sqrt(1+9*x*(2+3*(3-4*x)*x)))^(1/3)). - _Vaclav Kotesovec_, Apr 15 2024

%t CoefficientList[Series[(6*2^(1/3)*x + 2^(2/3)*(1 + 9*x + Sqrt[1 + 9*x*(2 + 3*(3 - 4*x)*x)])^(2/3))/(2*(1 + 9*x + Sqrt[1 + 9*x*(2 + 3*(3 - 4*x)*x)])^(1/3)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Apr 15 2024 *)

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

%Y Cf. A371988.

%K sign

%O 0,2

%A _Seiichi Manyama_, Apr 15 2024