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A168584
Number of ways of partitioning the multiset {1,1,2,3,...,n-1} into exactly four nonempty parts.
4
1, 7, 41, 215, 1041, 4767, 21001, 90055, 378881, 1572527, 6463161, 26375895, 107081521, 433076287, 1746588521, 7029269735, 28245956961, 113370724047, 454644109081, 1822061123575, 7298700653201, 29226175283807
OFFSET
4,2
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.
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)).
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 *)
PROG
(Magma) [(5*4^(n-3)-3^(n-1)+3*2^(n-2)-2)/3: n in [4..30]]; // Wesley Ivan Hurt, Dec 12 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Martin Griffiths, Nov 30 2009
STATUS
approved