%I #4 Nov 09 2019 16:28:53
%S 0,1,2,5,22,119,825,6810,65766,725139,8997795,124039530,1881019965,
%T 31117851270,557686108980,10763514011250,222577767068086,
%U 4909509776289707,115059754193953599,2855172351859669458,74786346248906702415,2062000166613319934190
%N Expansion of e.g.f. -log(1 - Sum_{k>=0} x^(2^k) / (2^k)!).
%F a(0) = 0; a(n) = A209229(n) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * A209229(n-k) * k * a(k).
%t nmax = 21; CoefficientList[Series[-Log[1 - Sum[x^(2^k)/(2^k)!, {k, 0, Floor[Log[2, nmax]] + 1}]], {x, 0, nmax}], x] Range[0, nmax]!
%t a[n_] := a[n] = Boole[IntegerQ[Log[2, n]]] + Sum[Binomial[n, k] Boole[IntegerQ[Log[2, n - k]]] k a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 21}]
%Y Cf. A000629, A115625, A209229.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Nov 09 2019