A333881 Expansion of e.g.f. exp(Sum_{k>=0} x^(3*k + 1) / (3*k + 1)!).

1, 1, 1, 1, 2, 6, 16, 37, 114, 478, 1907, 6777, 28414, 148579, 758916, 3580189, 18981485, 117883917, 720627553, 4193077474, 26795418840, 191751387094, 1352954503595, 9301704998742, 69285817230370, 559142785301527, 4453089770243547, 35182348161102172

Offset: 0

Author

Ilya Gutkovskiy, Apr 08 2020

Keywords

• nonn

Comments

Number of partitions of n-set into blocks congruent to 1 mod 3.

Crossrefs

Cf. A000110, A003724, A193437, A306347, A333882, A333883.

Programs

Mathematica

nmax = 27; CoefficientList[Series[Exp[Sum[x^(3 k + 1)/(3 k + 1)!, {k, 0, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
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}]
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 *)

Formula

E.g.f.: exp(exp(x)/3 - 2*sin(Pi/6 - sqrt(3)*x/2) / (3*exp(x/2))). - Vaclav Kotesovec, Apr 15 2020

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