A292798 - OEIS

%I #14 Sep 03 2024 15:06:16

%S 1,2,15,244,6345,229326,10663471,607693640,41058670113,3210853971610,

%T 285387481699551,28423216247375676,3136023698489382025,

%U 379743303818657805222,50074394496591697023135,7143088376895580682492176,1096075604718147681983312001,180030794404631168482202007090

%N a(n) = [x^n] 1/(1 - x - n*x/(1 - x - n*x/(1 - x - n*x/(1 - x - n*x/(1 - x - n*x/(1 - ...)))))), a continued fraction.

%F a(n) ~ exp(1/2) * 2^(2*n) * n^(n-3/2) / sqrt(Pi). - _Vaclav Kotesovec_, Sep 24 2017

%F a(n) = (1/(n+1)) * [x^n] (1+x)^(n+1) / (1 - n*x)^(n+1). - _Paul D. Hanna_, May 07 2018

%F From _Fabian Pereyra_, Sep 02 2024: (Start)

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

%F a(n) = [x^n] 2/(1 - x + sqrt(1 - 2*x*(1 + 2*n) + x^2)). (End)

%t Table[SeriesCoefficient[1/(1 - x + ContinuedFractionK[-n x, 1 - x, {i, 1, n}]), {x, 0, n}], {n, 0, 17}]

%t Table[SeriesCoefficient[(1 - x + Sqrt[1 - 2 (2 n + 1) x + x^2])/(1 - 2 (n + 1) x + x^2 - (x - 1) Sqrt[1 - 2 (2 n + 1) x + x^2]), {x, 0, n}], {n, 0, 17}]

%t Table[Hypergeometric2F1[-n, n + 1, 2, -n], {n, 0, 17}]

%o (PARI) {a(n) = polcoeff( (1+x)^(n+1) / (1 - n*x +x*O(x^n) )^(n+1), n) / (n+1)}

%o for(n=0,30,print1(a(n),", ")) \\ _Paul D. Hanna_, May 07 2018

%Y Main diagonal of A103209 (with offset 0).

%Y Cf. A006318.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Sep 23 2017