A292798 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.

1, 2, 15, 244, 6345, 229326, 10663471, 607693640, 41058670113, 3210853971610, 285387481699551, 28423216247375676, 3136023698489382025, 379743303818657805222, 50074394496591697023135, 7143088376895580682492176, 1096075604718147681983312001, 180030794404631168482202007090

Offset: 0

Author

Ilya Gutkovskiy, Sep 23 2017

Keywords

• nonn

Crossrefs

Main diagonal of A103209 (with offset 0).

Cf. A006318.

Programs

Mathematica

Table[SeriesCoefficient[1/(1 - x + ContinuedFractionK[-n x, 1 - x, {i, 1, n}]), {x, 0, n}], {n, 0, 17}]
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}]
Table[Hypergeometric2F1[-n, n + 1, 2, -n], {n, 0, 17}]

PARI

{a(n) = polcoeff( (1+x)^(n+1) / (1 - n*x +x*O(x^n) )^(n+1), n) / (n+1)}
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, May 07 2018

Formula

a(n) ~ exp(1/2) * 2^(2*n) * n^(n-3/2) / sqrt(Pi). - Vaclav Kotesovec, Sep 24 2017

a(n) = (1/(n+1)) * [x^n] (1+x)^(n+1) / (1 - n*x)^(n+1). - Paul D. Hanna, May 07 2018

From Fabian Pereyra, Sep 02 2024: (Start)

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

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