A346960 a(0) = 0, a(1) = 1; a(n) = n * (n+1) * a(n-1) + a(n-2).

0, 1, 6, 73, 1466, 44053, 1851692, 103738805, 7471045652, 672497847485, 73982234269002, 9766327421355749, 1523621059965765846, 277308799241190739721, 58236371461710021107256, 13977006459609646256481161, 3801803993385285491783983048, 1163365998982356970132155293849

Offset: 0

Author

Ilya Gutkovskiy, Aug 13 2021

Keywords

• nonn
• frac

Comments

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

Examples

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

Crossrefs

Cf. A001053, A002378, A036245, A058307, A071896, A347051, A347052.

Programs

Mathematica

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

Formula

a(n) ~ c * n^(2*n + 2) / exp(2*n), where c = 3.2100642122891047165999468271849715691225751316633504931782933233387646256... - Vaclav Kotesovec, Aug 14 2021