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%I #13 May 11 2020 22:38:11
%S 1,1,0,1,4,0,1,7,0,84,841,11,0,286,4004,1,8024,136136,816,7775256,
%T 155195040,54265,1193830,0,109832360,2749077760,84987760,296010,
%U 10716746041,310545275069,1201800600,2444026056820,77016647623040,0,14402113079955304,504073957798435640
%N E.g.f.: Product_{k>=1} (1 + x^(k*(k + 1)/2) / (k*(k + 1)/2)!).
%t nmax = 35; CoefficientList[Series[Product[(1 + x^(k (k + 1)/2)/(k (k + 1)/2)!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[DivisorSum[k, -#/(-#!)^(k/#) &, IntegerQ[Sqrt[8 # + 1]] &] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 35}]
%Y Cf. A007837, A032310, A053614 (positions of 0's), A115278, A205799.
%K nonn
%O 0,5
%A _Ilya Gutkovskiy_, May 11 2020
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