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MATHEMATICA
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nmax = 23; CoefficientList[Series[Product[1/(1 - x^2^k)^(1/2^k), {k, 0, Ceiling[Log[2, nmax]]}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Exp[Sum[(IntegerExponent[k, 2] + 1) x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (n - 1)! Sum[(IntegerExponent[k, 2] + 1) a[n - k]/(n - k)!, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]
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