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%I #8 Mar 10 2020 04:42:04
%S 1,1,3,8,20,50,121,297,716,1739,4198,10157,24513,59246,143006,345381,
%T 833792,2013272,4860337,11734717,28329772,68396030,165121957,
%U 398644144,962410246,2323475153,5609360573,13542220814,32693802921,78929886033,190553574988,460037180829,1110627936647
%N a(1) = 1; a(n+1) = Sum_{k=1..n} a(k) * ceiling(n/k).
%F G.f. A(x) satisfies: A(x) = x * (1 + (1/(1 - x)) * (A(x) + x * Sum_{k>=1} A(x^k))).
%F a(1) = 1; a(n) = a(n-1) + Sum_{k=1..n-2} (a(k) + Sum_{d|k} a(d)).
%F a(n) ~ c * (1 + sqrt(2))^n, where c = 0.2594006517235012546870541901936538347053403598092060748627156661727... - _Vaclav Kotesovec_, Mar 10 2020
%t a[1] = 1; a[n_] := a[n] = Sum[a[k] Ceiling[(n - 1)/k], {k, 1, n - 1}]; Table[a[n], {n, 1, 33}]
%t a[1] = 1; a[n_] := a[n] = a[n - 1] + Sum[a[k] + Sum[a[d], {d, Divisors[k]}], {k, 1, n - 2}]; Table[a[n], {n, 1, 33}]
%t terms = 33; A[_] = 0; Do[A[x_] = x (1 + (1/(1 - x)) (A[x] + x Sum[A[x^k], {k, 1, terms}])) + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest
%Y Cf. A006590, A014668, A097919, A332490.
%K nonn
%O 1,3
%A _Ilya Gutkovskiy_, Feb 26 2020
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