A347051 a(0) = 1, a(1) = 2; a(n) = n * (n+1) * a(n-1) + a(n-2).

1, 2, 13, 158, 3173, 95348, 4007789, 224531532, 16170278093, 1455549559902, 160126621867313, 21138169636045218, 3297714589844921321, 600205193521411725640, 126046388354086307305721, 30251733410174235165098680, 8228597533955746051214146681, 2517981097123868465906693983066

Offset: 0

Author

Ilya Gutkovskiy, Aug 13 2021

Keywords

• nonn
• frac

Comments

a(n) is the denominator of fraction equal to the continued fraction [0; 2, 6, 12, 20, 30, ..., n*(n+1)].

Examples

			a(1) =    2 because 1/(1*2)                               = 1/2.
a(2) =   13 because 1/(1*2 + 1/(2*3))                     = 6/13.
a(3) =  158 because 1/(1*2 + 1/(2*3 + 1/(3*4)))           = 73/158.
a(4) = 3173 because 1/(1*2 + 1/(2*3 + 1/(3*4 + 1/(4*5)))) = 1466/3173.
		

Crossrefs

Cf. A001040, A001046, A002378, A036246, A071896, A102038, A346960, A347052.

Programs

Mathematica

a[0] = 1; a[1] = 2; a[n_] := a[n] = n (n + 1) a[n - 1] + a[n - 2]; Table[a[n], {n, 0, 17}]
Table[Denominator[ContinuedFractionK[1, k (k + 1), {k, 1, n}]], {n, 0, 17}]

Formula

a(n) ~ c * n^(2*n + 2) / exp(2*n), where c = 6.9478401587876967481571909904361736371398357108358019737901443045685048723... - Vaclav Kotesovec, Aug 14 2021