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%I #16 Aug 14 2021 15:02:34
%S 0,1,6,73,1466,44053,1851692,103738805,7471045652,672497847485,
%T 73982234269002,9766327421355749,1523621059965765846,
%U 277308799241190739721,58236371461710021107256,13977006459609646256481161,3801803993385285491783983048,1163365998982356970132155293849
%N a(0) = 0, a(1) = 1; a(n) = n * (n+1) * a(n-1) + a(n-2).
%C a(n) is the numerator 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 = 3.2100642122891047165999468271849715691225751316633504931782933233387646256... - _Vaclav Kotesovec_, Aug 14 2021
%e a(1) = 1 because 1/(1*2) = 1/2.
%e a(2) = 6 because 1/(1*2 + 1/(2*3)) = 6/13.
%e a(3) = 73 because 1/(1*2 + 1/(2*3 + 1/(3*4))) = 73/158.
%e a(4) = 1466 because 1/(1*2 + 1/(2*3 + 1/(3*4 + 1/(4*5)))) = 1466/3173.
%t 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}]
%t Table[Numerator[ContinuedFractionK[1, k (k + 1), {k, 1, n}]], {n, 0, 17}]
%Y Cf. A001053, A002378, A036245, A058307, A071896, A347051, A347052.
%K nonn,frac
%O 0,3
%A _Ilya Gutkovskiy_, Aug 13 2021
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