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MATHEMATICA
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nmax = 15; CoefficientList[Series[Exp[BesselI[0, 2 Sqrt[x]]^4 - 1], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 Binomial[2 k, k] HypergeometricPFQ[{1/2, -k, -k, -k}, {1, 1, 1/2 - k}, 1] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]
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