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a(0) = 1; a(n) = (1/n) * Sum_{k=3..n} (binomial(n,k) * k!)^2 * a(n-k) / k.
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%I #5 Jul 13 2021 09:19:01

%S 1,0,0,4,36,576,17600,694800,35802144,2391438336,200018045952,

%T 20476348214400,2521840589347200,368057828019898368,

%U 62841061478699292672,12413136137144581203456,2809529229255558769612800,722458985698006017844838400,209487621780682072569567903744

%N a(0) = 1; a(n) = (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 - x^2 / 4 ).

%F Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( Sum_{n>=3} x^n / n^2 ).

%t a[0] = 1; a[n_] := a[n] = (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 - x^2/4], {x, 0, nmax}], x] Range[0, nmax]!^2

%Y Cf. A038205, A074707, A346291.

%K nonn

%O 0,4

%A _Ilya Gutkovskiy_, Jul 13 2021