A333883 Expansion of e.g.f. exp(Sum_{k>=0} x^(6*k + 1) / (6*k + 1)!).

1, 1, 1, 1, 1, 1, 1, 2, 9, 37, 121, 331, 793, 1718, 5163, 32281, 217921, 1188709, 5291353, 20031170, 66744741, 267996541, 2030569465, 18368560519, 138812739409, 853152218102, 4409607501927, 19826125988257, 99717123889777, 871344991322017, 9658479225877057

Offset: 0

Author

Ilya Gutkovskiy, Apr 08 2020

Keywords

• nonn

Comments

Number of partitions of n-set into blocks congruent to 1 mod 6.

Crossrefs

Cf. A000110, A003724, A306347, A333881, A333882.

Programs

Mathematica

nmax = 30; CoefficientList[Series[Exp[Sum[x^(6 k + 1)/(6 k + 1)!, {k, 0, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Boole[MemberQ[{1}, Mod[k, 6]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 30}]
nmax = 30; CoefficientList[Series[Exp[x*HypergeometricPFQ[{}, {1/3, 1/2, 2/3, 5/6, 7/6}, x^6/46656]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 15 2020 *)

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/6)} binomial(n-1,6*k) * a(n-6*k-1). - Seiichi Manyama, Sep 22 2023