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%I #8 Aug 14 2021 15:02:28
%S 1,2,13,158,3173,95348,4007789,224531532,16170278093,1455549559902,
%T 160126621867313,21138169636045218,3297714589844921321,
%U 600205193521411725640,126046388354086307305721,30251733410174235165098680,8228597533955746051214146681,2517981097123868465906693983066
%N a(0) = 1, a(1) = 2; a(n) = n * (n+1) * a(n-1) + a(n-2).
%C a(n) is the denominator of fraction equal to the continued fraction [0; 2, 6, 12, 20, 30, ..., n*(n+1)].
%F a(n) ~ c * n^(2*n + 2) / exp(2*n), where c = 6.9478401587876967481571909904361736371398357108358019737901443045685048723... - _Vaclav Kotesovec_, Aug 14 2021
%e a(1) = 2 because 1/(1*2) = 1/2.
%e a(2) = 13 because 1/(1*2 + 1/(2*3)) = 6/13.
%e a(3) = 158 because 1/(1*2 + 1/(2*3 + 1/(3*4))) = 73/158.
%e a(4) = 3173 because 1/(1*2 + 1/(2*3 + 1/(3*4 + 1/(4*5)))) = 1466/3173.
%t 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}]
%t Table[Denominator[ContinuedFractionK[1, k (k + 1), {k, 1, n}]], {n, 0, 17}]
%Y Cf. A001040, A001046, A002378, A036246, A071896, A102038, A346960, A347052.
%K nonn,frac
%O 0,2
%A _Ilya Gutkovskiy_, Aug 13 2021