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%I #87 Jan 27 2024 07:11:10
%S 1,8,60,448,3360,25344,192192,1464320,11202048,85995520,662165504,
%T 5112102912,39557939200,306726174720,2382605107200,18537602088960,
%U 144438816276480,1126891074355200,8802271391907840,68829791335219200,538749548542033920,4220762508660572160
%N a(n) = 2^n*binomial(2*(n+1), n).
%C The g.f. is the derivative of the REVERT transform of x/(1+2*x)^2. - _Thomas Baruchel_, Jul 02 2018
%H G. C. Greubel, <a href="/A245391/b245391.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = 2^(3*n+2)*(n+1/2)!/(sqrt(Pi)*(n+2)*n!).
%F a(n) = (4*(2*n+1)*(n+1))/(n*(n+2))*a(n-1) for n >= 1.
%F O.g.f.: 4/(sqrt(1-8*x)*(1+sqrt(1-8*x))^2).
%F a(n) = A000079(n)*A001791(n+1). - _Robert G. Wilson v_, Aug 08 2018
%F From _Amiram Eldar_, Jan 27 2024: (Start)
%F Sum_{n>=0} 1/a(n) = 2/7 + 44*arccot(sqrt(7))/(7*sqrt(7)).
%F Sum_{n>=0} (-1)^n/a(n) = 2/9 + 26*log(2)/27. (End)
%p a := n -> 2^(3*n+2)*(n+1/2)!/(sqrt(Pi)*(n+2)*n!):
%p seq(a(n), n = 0..21);
%t CoefficientList[Series[4/(Sqrt[1 - 8*x]*(1 + Sqrt[1 - 8*x])^2), {x,0, 50}], x] (* _G. C. Greubel_, Apr 06 2017 *)
%t a[n_] := 2^n*Binomial[2 n + 2, n]; Array[a, 22, 0] (* _Robert G. Wilson v_, Aug 08 2018 *)
%o (Sage)
%o @CachedFunction
%o def A245391(n):
%o return (4*(2*n+1)*(n+1))/(n*(n+2))*a(n-1) if n > 0 else 1
%o [A245391(n) for n in range(22)]
%o (PARI) my(x='x+O('x^50)); Vec(4/(sqrt(1-8*x)*(1+sqrt(1-8*x))^2)) \\ _G. C. Greubel_, Apr 06 2017
%o (PARI) my(x='x+O('x^33)); Vec(deriv(serreverse(x/(1+2*x)^2))) \\ _Thomas Baruchel_, Jul 02 2018
%Y Cf. A000079, A001791.
%K nonn
%O 0,2
%A _Peter Luschny_, Nov 30 2014
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