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

Offset: 3

Author

Martin Griffiths, Dec 01 2009

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A168604, A168606.

Programs

Magma

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

Mathematica

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

Sage

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

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)).

Extensions

Last element of the multiset in the definition corrected by Martin Griffiths, Dec 02 2009