A168605 - OEIS

A168605

Number of ways of partitioning the multiset {1,1,1,2,3,...,n-2} into exactly three nonempty parts.

1, 2, 8, 30, 104, 342, 1088, 3390, 10424, 31782, 96368, 291150, 877544, 2640822, 7938848, 23849310, 71613464, 214971462, 645176528, 1936053870, 5809210184, 17429727702, 52293377408, 156888520830, 470682339704, 1412080573542

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OFFSET

3,2

COMMENTS

The number of ways of partitioning the multiset {1, 1, 1, 2, 3, ..., n-1} into exactly two and four nonempty parts are given in A168604 and A168606, respectively.

LINKS

G. C. Greubel, Table of n, a(n) for n = 3..1000

M. Griffiths and I. Mezo, A generalization of Stirling Numbers of the Second Kind via a special multiset, JIS 13 (2010) #10.2.5.

Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

FORMULA

a(n) = (5*3^(n-3) - 3*2^(n-2) + 3)/3 for n >= 4, with a(3) = 1.

The shifted e.g.f. is (5*exp(3*x) - 6*exp(2*x) + 3*exp(x) + 1)/3.

G.f.: x^3*(1 -4*x +7*x^2 -2*x^3)/((1-x)*(1-2*x)*(1-3*x)).

MATHEMATICA

a[n_]:= If[n==3, 1, (5*3^(n-3) - 3*2^(n-2) + 3)/3]; Table[a[n], {n, 3, 30}]

PROG

(SageMath) [1]+[(5*3^(n-3) -3*2^(n-2) +3)/3 for n in (4..30)] # G. C. Greubel, Feb 07 2021