A168585 -id:A168585 - cp's OEIS frontend

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

Martin Griffiths, Nov 30 2009

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.

			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}}.

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).

Cf. A083329, A126644, A168584, A168585.

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

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

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

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

A241500 Triangle T(n,k): number of ways of partitioning the n-element multiset {1,1,2,3,...,n-1} into exactly k nonempty parts, n>=1 and 1<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 4, 1, 1, 11, 16, 7, 1, 1, 23, 58, 41, 11, 1, 1, 47, 196, 215, 90, 16, 1, 1, 95, 634, 1041, 640, 176, 22, 1, 1, 191, 1996, 4767, 4151, 1631, 315, 29, 1, 1, 383, 6178, 21001, 25221, 13587, 3696, 526, 37, 1, 1, 767, 18916, 90055, 146140, 105042, 38409, 7638, 831, 46, 1

Offset: 1

Andrew Woods, Apr 24 2014

			There are 58 ways to partition {1,1,2,3,4,5} into three nonempty parts.
The first few rows are:
  1;
  1,   1;
  1,   2,    1;
  1,   5,    4,     1;
  1,  11,   16,     7,     1;
  1,  23,   58,    41,    11,     1;
  1,  47,  196,   215,    90,    16,    1;
  1,  95,  634,  1041,   640,   176,   22,   1;
  1, 191, 1996,  4767,  4151,  1631,  315,  29,  1;
  1, 383, 6178, 21001, 25221, 13587, 3696, 526, 37, 1;
  ...

The first five columns appear as A000012, A083329, A168583, A168584, A168585.

Row sums give A035098.

T(n,k) = stirling(n-1,k,2) + stirling(n-1,k-1,2) + binomial(k,2)*stirling(n-2,k,2); \\ Michel Marcus, Apr 24 2014

T(n,k) = S(n-1,k) + S(n-1,k-1) + C(k,2)*S(n-2,k), where S refers to Stirling numbers of the second kind (A008277), and C to binomial coefficients (A007318).

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

Original entry on oeis.org

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

Offset: 4

Martin Griffiths, Nov 30 2009

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.

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).

Cf. A083329, A168583, A168585.

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

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

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 *)

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)).

cp's OEIS Frontend

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

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A241500 Triangle T(n,k): number of ways of partitioning the n-element multiset {1,1,2,3,...,n-1} into exactly k nonempty parts, n>=1 and 1<=k<=n.

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

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Magma

Maple

Mathematica

Formula