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%I #14 Sep 22 2023 12:18:28
%S 1,1,1,1,2,6,16,37,114,478,1907,6777,28414,148579,758916,3580189,
%T 18981485,117883917,720627553,4193077474,26795418840,191751387094,
%U 1352954503595,9301704998742,69285817230370,559142785301527,4453089770243547,35182348161102172
%N Expansion of e.g.f. exp(Sum_{k>=0} x^(3*k + 1) / (3*k + 1)!).
%C Number of partitions of n-set into blocks congruent to 1 mod 3.
%F E.g.f.: exp(exp(x)/3 - 2*sin(Pi/6 - sqrt(3)*x/2) / (3*exp(x/2))). - _Vaclav Kotesovec_, Apr 15 2020
%F a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n-1,3*k) * a(n-3*k-1). - _Seiichi Manyama_, Sep 22 2023
%t nmax = 27; CoefficientList[Series[Exp[Sum[x^(3 k + 1)/(3 k + 1)!, {k, 0, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t a[n_] := a[n] = Sum[Boole[MemberQ[{1}, Mod[k, 3]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 27}]
%t nmax = 30; CoefficientList[Series[Exp[Exp[x]/3 - 2*Sin[Pi/6 - Sqrt[3]*x/2] / (3*Exp[x/2])], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Apr 15 2020 *)
%Y Cf. A000110, A003724, A193437, A306347, A333882, A333883.
%K nonn
%O 0,5
%A _Ilya Gutkovskiy_, Apr 08 2020
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