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%I #5 Jul 14 2021 14:55:37
%S 1,1,2,10,124,2396,64856,2452472,124483360,8146185504,668645524032,
%T 67374446014272,8183368905811584,1179807474740449920,
%U 199266648878034317568,38984601149045449948416,8748103140554862876727296,2232274640259371687436982272,642805438643602793466093711360
%N a(0) = 1; a(n) = n * a(n-1) + (1/n) * Sum_{k=3..n} (binomial(n,k) * k!)^2 * a(n-k) / k.
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( polylog(2,x) - x^2 / 4 ).
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( x + Sum_{n>=3} x^n / n^2 ).
%t a[0] = 1; a[n_] := a[n] = n a[n - 1] + (1/n) Sum[(Binomial[n, k] k!)^2 a[n - k]/k, {k, 3, n}]; Table[a[n], {n, 0, 18}]
%t nmax = 18; CoefficientList[Series[Exp[PolyLog[2, x] - x^2/4], {x, 0, nmax}], x] Range[0, nmax]!^2
%Y Cf. A000266, A074707, A346291.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jul 14 2021
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