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%I #6 Jul 12 2020 12:18:00
%S 1,-1,1,2,-15,-46,880,5837,-132783,-2109238,35966256,1440196097,
%T -8909037720,-1504006716551,-16097564749643,2021100230840147,
%U 83130656159529937,-2475528081920694566,-331363460045748820376,-3874344448291316066455,1255007424437046915956520
%N a(0) = 1; a(n) = -(1/n) * Sum_{k=0..n-1} binomial(n,k)^2 * (n-k) * a(k).
%F a(n) = (n!)^2 * [x^n] exp(-Sum_{k>=1} x^k / (k!)^2).
%F a(n) = (n!)^2 * [x^n] exp(1 - BesselI(0,2*sqrt(x))).
%t a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Binomial[n, k]^2 (n - k) a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 20}]
%t nmax = 20; CoefficientList[Series[Exp[-Sum[x^k/(k!)^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2
%Y Cf. A000587, A023998, A336210.
%K sign
%O 0,4
%A _Ilya Gutkovskiy_, Jul 12 2020
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