A168583 The number of ways of partitioning the multiset {1,1,2,3,...,n-1} into exactly three nonempty parts.

1, 4, 16, 58, 196, 634, 1996, 6178, 18916, 57514, 174076, 525298, 1582036, 4758394, 14299756, 42948418, 128943556, 387027274, 1161475036, 3485211538, 10457207476, 31374768154, 94130595916, 282404370658, 847238277796, 2541765165034, 7625396158396

Offset

3,2

Comments

The number of ways of partitioning the multiset {1, 1, 2, 3, ..., n-1} into exactly two, four and five nonempty parts are given in A083329, A168584 and A168585, respectively.

Links

Table of n, a(n) for n=3..29.

M. Griffiths, 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

For a>=3, a(n) = 3^(n-2) - 3*2^(n-3) + 1.

E.g.f.: 3*e^(3x) - 3*e^(2x) + e^x (shifted).

O.g.f.: x^3*(1-2x+3x^2)/((1-x)*(1-2x)*(1-3x)).

a(n) = A126644(n-3). - R. J. Mathar, Dec 11 2009

Example

The partitions of {1,1,2,3} into exactly three nonempty parts are {{1},{1},{2,3}}, {{1},{2},{1,3}}, {{1},{3},{1,2}} and {{2},{3},{1,1}}.

Maple

A168583:=n->3^(n-2)-3*2^(n-3)+1: seq(A168583(n), n=3..40); # Wesley Ivan Hurt, Dec 12 2015

Mathematica

f1[n_] := 3^(n - 2) - 3 2^(n - 3) + 1; Table[f1[n], {n, 3, 25}]

Prog

(Magma) [3^(n-2) - 3*2^(n-3) + 1: n in [3..35]]; // Vincenzo Librandi, Dec 12 2015

Crossrefs

Cf. A083329, A126644, A168584, A168585.

Sequence in context: A034514 A263851 A126644 * A092688 A267466 A255299

Adjacent sequences: A168580 A168581 A168582 * A168584 A168585 A168586

Keyword

nonn,easy

Author

Martin Griffiths, Nov 30 2009

Status

approved