A168584 Number of ways of partitioning the multiset {1,1,2,3,...,n-1} into exactly four nonempty parts.

1, 7, 41, 215, 1041, 4767, 21001, 90055, 378881, 1572527, 6463161, 26375895, 107081521, 433076287, 1746588521, 7029269735, 28245956961, 113370724047, 454644109081, 1822061123575, 7298700653201, 29226175283807

Offset: 4

Author

Martin Griffiths, Nov 30 2009

Keywords

• nonn
• easy

Comments

The number of ways of partitioning the multiset {1, 1, 2, 3, ..., n-1} into exactly two, three and five nonempty parts are given in A083329, A168583 and A168585, respectively.

Links

• 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 (10,-35,50,-24).

Crossrefs

Cf. A083329, A168583, A168585.

Programs

Magma

[(5*4^(n-3)-3^(n-1)+3*2^(n-2)-2)/3: n in [4..30]]; // Wesley Ivan Hurt, Dec 12 2015

Maple

A168584:=n->(5*4^(n-3)-3^(n-1)+3*2^(n-2)-2)/3: seq(A168584(n), n=4..30); # Wesley Ivan Hurt, Dec 12 2015

Mathematica

f2[n_] := 1/3 (5 4^(n - 3) - 3^(n - 1) + 3 2^(n - 2) - 2); Table[f2[n], {n, 4, 25}]
LinearRecurrence[{10,-35,50,-24},{1,7,41,215},30] (* Harvey P. Dale, Sep 15 2020 *)

Formula

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

The shifted exponential generating function is (20e^(4x) - 27e^(3x) + 12e^(2x) - 2e^x)/3.

The ordinary generating function is x^4(1-3x+6x^2)/((1-x)(1-2x)(1-3x)(1-4x)).