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%I #6 Nov 09 2019 16:28:41
%S 0,1,1,3,10,44,251,1707,13496,122108,1243201,14060771,174932274,
%T 2374268974,34910039164,552782630401,9378254813944,169714311278784,
%U 3263200704705648,66434349885323328,1427653109477475098,32294539445483981821,767051977023372086530
%N Expansion of e.g.f. -log(1 - Sum_{k>=1} x^(k*(k + 1)/2) / (k*(k + 1)/2)!).
%F a(0) = 0; a(n) = A010054(n) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * A010054(n-k) * k * a(k).
%t nmax = 22; CoefficientList[Series[-Log[1 - Sum[x^(k (k + 1)/2)/(k (k + 1)/2)!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t a[n_] := a[n] = Boole[IntegerQ[(8 n + 1)^(1/2)]] + Sum[Binomial[n, k] Boole[IntegerQ[(8 (n - k) + 1)^(1/2)]] k a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 22}]
%Y Cf. A000629, A010054, A205803, A329259.
%K nonn
%O 0,4
%A _Ilya Gutkovskiy_, Nov 09 2019
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