A168605 -id:A168605 - cp's OEIS frontend

A168604 a(n) = 2^(n-2) - 1.

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 16777215, 33554431, 67108863, 134217727, 268435455, 536870911, 1073741823, 2147483647, 4294967295, 8589934591

Offset: 3

Martin Griffiths, Dec 01 2009

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

An elephant sequence, see A175655. For the central square six A[5] vectors, with decimal values between 26 and 176, lead to this sequence. For the corner squares these vectors lead to the companion sequence A000325 (without the first leading 1). - Johannes W. Meijer, Aug 15 2010

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

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
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 (3,-2).

The number of ways of partitioning the multiset {1, 1, 1, 2, 3, ..., n-1} into exactly three and four nonempty parts are given in A168605 and A168606, respectively.

[2^(n-2)-1 : n in [3..35]]; // Vincenzo Librandi, May 13 2011

f4[n_] := 2^(n - 2) - 1; Table[f4[n], {n, 3, 30}]
LinearRecurrence[{3,-2},{1,3},40] (* Harvey P. Dale, Oct 20 2013 *)

a(n)=2^(n-2)-1 \\ Charles R Greathouse IV, Oct 07 2015

E.g.f.: 2*exp(2*x)-exp(x).
a(n) = A000225(n-2).
G.f.: x^3/((1-x)*(1-2*x))
a(n) = A126646(n-3). - R. J. Mathar, Dec 11 2009
a(n) = 3*a(n-1) - 2*a(n-2). - Arkadiusz Wesolowski, Jun 14 2013
a(n) = A000918(n-2) + 1. - Miquel Cerda, Aug 09 2016

A291117 Triangle read by rows: T(n,k) = number of ways of partitioning the (n+2)-element multiset {1,1,1,2,3,...,n} into exactly k nonempty parts, n >= 0 and 1 <= k <= n + 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 7, 8, 4, 1, 1, 15, 30, 20, 7, 1, 1, 31, 104, 102, 46, 11, 1, 1, 63, 342, 496, 300, 96, 16, 1, 1, 127, 1088, 2294, 1891, 786, 183, 22, 1, 1, 255, 3390, 10200, 11417, 6167, 1862, 323, 29, 1, 1, 511, 10424, 44062, 66256, 46417, 17801, 4040, 535, 37, 1, 1, 1023, 31782, 186416, 372190, 336022, 162372, 46425, 8127, 841, 46, 1

Offset: 0

Marko Riedel, Aug 17 2017

			Triangle begins:
  1,   1;
  1,   1,   1;
  1,   3,   2,   1;
  1,   7,   8,   4,   1;
  1,  15,  30,  20,   7,  1;
  1,  31, 104, 102,  46, 11,  1;
  1,  63, 342, 496, 300, 96, 16, 1;

M. Griffiths and I. Mezo, A generalization of Stirling Numbers of the Second Kind via a special multiset, JIS 13 (2010) #10.2.5.
Marko Riedel, Partitions into bounded blocks, Mathematics Stack Exchange.
Marko Riedel, Maple code for sequences A241500, A291117, A291118, A291119, A291120.

Cf. A241500, A291118, A291119, A291120.

Columns k=1..4: A000012, A255047, A168605, A168606.

Formula including proof is at web link.

A168606 The number of ways of partitioning the multiset {1,1,1,2,3,...,n-2} into exactly four nonempty parts.

Original entry on oeis.org

1, 4, 20, 102, 496, 2294, 10200, 44062, 186416, 776934, 3203080, 13101422, 53279136, 215749174, 870919160, 3507493182, 14101520656, 56620923014, 227128606440, 910449955342, 3647607982976, 14607859562454, 58483727432920

Offset: 4

Martin Griffiths, Dec 01 2009

The number of ways of partitioning the multiset {1, 1, 1, 2, 3, ..., n-1} into exactly two and three nonempty parts are given in A168604 and A168605 respectively.

G. C. Greubel, Table of n, a(n) for n = 4..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 (10,-35,50,-24).

Cf. A168604, A168605.

[(10*4^(n-4) -5*3^(n-3) +9*2^(n-4) -1)/3: n in [4..30]]; // G. C. Greubel, Feb 07 2021

a[n_]:= (10*4^(n-4) - 5*3^(n-3) + 9*2^(n-4) - 1)/3; Table[a[n], {n, 4, 30}]

[(10*4^(n-4) -5*3^(n-3) +9*2^(n-4) -1)/3 for n in (4..30)] # G. C. Greubel, Feb 07 2021

a(n) = (10*4^(n-4) - 5*3^(n-3) + 9*2^(n-4) - 1)/3.
The shifted e.g.f. is (10*exp(4*x) - 15*exp(3*x) + 9*exp(2*x) - exp(x))/3.
G.f.: x^4*(1 -6*x +15*x^2 -8*x^3)/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)).

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

cp's OEIS Frontend

A168604 a(n) = 2^(n-2) - 1.

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A291117 Triangle read by rows: T(n,k) = number of ways of partitioning the (n+2)-element multiset {1,1,1,2,3,...,n} into exactly k nonempty parts, n >= 0 and 1 <= k <= n + 2.

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A168606 The number of ways of partitioning the multiset {1,1,1,2,3,...,n-2} into exactly four nonempty parts.

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