A241500 -id:A241500 - cp's OEIS frontend

A035098 Near-Bell numbers: partitions of an n-multiset with multiplicities 1, 1, 1, ..., 1, 2.

1, 2, 4, 11, 36, 135, 566, 2610, 13082, 70631, 407846, 2504071, 16268302, 111378678, 800751152, 6027000007, 47363985248, 387710909055, 3298841940510, 29119488623294, 266213358298590, 2516654856419723, 24566795704844210

Offset: 1

George Beck

nonn

A035098 and A000070 are near the two ends of a spectrum. Another way to look at A000070 is as the number of partitions of an n-multiset with multiplicities n-1, 1.
The very ends are the number of partitions and the Stirling numbers of the second kind, which count the n-multiset partitions with multiplicities n and 1,1,1,...,1, respectively.
Intermediate sequences are the number of ways of partitioning an n-multiset with multiplicities some partition of n.

			a(3)=4 because there are 4 ways to partition the multiset {1,2,2} (with multiplicities {1,2}): {{1,2,2}} {{1,2},{2}} {{1},{2,2}} {{1},{2},{2}}.

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778. - N. J. A. Sloane, Dec 30 2018

Alois P. Heinz, Table of n, a(n) for n = 1..576 (first 200 terms from Vincenzo Librandi)
M. Griffiths, Generalized Near-Bell Numbers, JIS 12 (2009) 09.5.7.
M. Griffiths, I. Mezo, A generalization of Stirling Numbers of the Second Kind via a special multiset, JIS 13 (2010) #10.2.5.
Martin Griffiths, Generating Functions for Extended Stirling Numbers of the First Kind, Journal of Integer Sequences, 17 (2014), #14.6.4.

Cf. A000070, A000110, A059606, A087648.
Row sums of A241500.
Column 1 of array in A322765.
Row n=2 of A346426.

with(combinat): a:= n-> floor(1/2*(bell(n-2)+bell(n-1)+bell(n))): seq(a(n), n=1..25); # Zerinvary Lajos, Oct 07 2007

f[n_] := Sum[ StirlingS2[n, k] ((k + 1) (k + 2)/2 + 1), {k, 0, n}]; Array[f, 22, 0]
f[n_] := (BellB[n] + BellB[n + 1] + BellB[n + 2])/2; Array[f, 22, 0]
Range[0, 21]! CoefficientList[ Series[ (1 + Exp@ x)^2/2 Exp[ Exp@ x - 1], {x, 0, 21}], x] (* 3 variants by Robert G. Wilson v, Jan 13 2011 *)
Join[{1},Total[#]/2&/@Partition[BellB[Range[0,30]],3,1]] (* Harvey P. Dale, Jan 02 2019 *)

Sum_{k=0..n} Stirling2(n, k)*((k+1)*(k+2)/2+1). E.g.f.: 1/2*(1+exp(x))^2*exp(exp(x)-1). (1/2)*(Bell(n)+Bell(n+1)+Bell(n+2)). - Vladeta Jovovic, Sep 23 2003 [for offset -1]

a(n) ~ Bell(n)/2 * (1 + LambertW(n)/n). - Vaclav Kotesovec, Jul 28 2021

More terms from Vladeta Jovovic, Sep 23 2003

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.

A291118 Triangle read by rows: T(n,k) number of ways of partitioning the (n+3)-element multiset {1,1,1,1,2,3,...,n} into exactly k nonempty parts, n >= 0 and 1 <= k <= n + 3.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 4, 4, 2, 1, 1, 9, 14, 9, 4, 1, 1, 19, 49, 43, 21, 7, 1, 1, 39, 164, 206, 123, 47, 11, 1, 1, 79, 529, 957, 741, 331, 97, 16, 1, 1, 159, 1664, 4294, 4409, 2397, 829, 184, 22, 1, 1, 319, 5149, 18713, 25551, 17353, 7105, 1919, 324, 29, 1, 1, 639, 15764, 79746, 144043, 123523, 60795, 19405, 4113, 536, 37, 1, 1, 1279, 47929, 334237, 792561, 859327, 514081, 193530, 49002, 8218, 842, 46, 1

Offset: 0

Marko Riedel, Aug 17 2017

			Triangle begins:
1,   1,    1;
1,   2,    1,    1;
1,   4,    4,    2,    1;
1,   9,   14,    9,    4,    1;
1,  19,   49,   43,   21,    7,   1;
1,  39,  164,  206,  123,   47,  11,   1;
1,  79,  529,  957,  741,  331,  97,  16,  1;
1, 159, 1664, 4294, 4409, 2397, 829, 184, 22, 1;

M. Griffiths, 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.

Cf. A241500, A291117, A291119, A291120.

Formula including proof is at web link.

A291119 Triangle read by rows: T(n,k) number of ways of partitioning the (n+4)-element multiset {1,1,1,1,1,2,3,...,n} into exactly k nonempty parts, n >= 0 and 1 <= k <= n + 4.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 5, 6, 4, 2, 1, 1, 11, 21, 17, 9, 4, 1, 1, 23, 72, 78, 47, 21, 7, 1, 1, 47, 237, 361, 265, 128, 47, 11, 1, 1, 95, 756, 1634, 1533, 847, 337, 97, 16, 1, 1, 191, 2361, 7197, 8819, 5826, 2571, 836, 184, 22, 1, 1, 383, 7272, 30958, 49807, 40433, 20419, 7372, 1927, 324, 29, 1, 1, 767, 22197, 130721, 275445, 278380, 165103, 66809, 19794, 4122, 536, 37, 1

Offset: 0

Marko Riedel, Aug 17 2017

			Triangle begins:
1,  2,   1,    1;
1,  2,   2,    1,    1;
1,  5,   6,    4,    2,   1;
1, 11,  21,   17,    9,   4,   1;
1, 23,  72,   78,   47,  21,   7,  1;
1, 47, 237,  361,  265, 128,  47, 11,  1;
1, 95, 756, 1634, 1533, 847, 337, 97, 16, 1;

M. Griffiths, 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.

Cf. A241500, A291117, A291118, A291120.

Formula including proof is at web link.

A291120 Triangle read by rows: T(n,k) number of ways of partitioning the (n+5)-element multiset {1,1,1,1,1,1,2,3,...,n} into exactly k nonempty parts, n >= 0 and 1 <= k <= n + 5.

Original entry on oeis.org

1, 2, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 6, 9, 7, 4, 2, 1, 1, 13, 30, 29, 18, 9, 4, 1, 1, 27, 100, 129, 92, 48, 21, 7, 1, 1, 55, 324, 581, 504, 287, 129, 47, 11, 1, 1, 111, 1024, 2577, 2834, 1844, 879, 338, 97, 16, 1, 1, 223, 3180, 11189, 15918, 12301, 6431, 2615, 837, 184, 22, 1, 1, 447, 9760, 47649, 88232, 83050, 49197, 21498, 7430, 1928, 324, 29, 1, 1, 895, 29724, 199781, 481044, 558819, 384913, 184823, 68606, 19868, 4123, 536, 37, 1

Offset: 0

Marko Riedel, Aug 17 2017

			Triangle begins:
1,   2,    2,    1,    1;
1,   3,    3,    2,    1,    1;
1,   6,    9,    7,    4,    2,   1;
1,  13,   30,   29,   18,    9,   4,   1;
1,  27,  100,  129,   92,   48,  21,   7,  1;
1,  55,  324,  581,  504,  287, 129,  47, 11,  1;
1, 111, 1024, 2577, 2834, 1844, 879, 338, 97, 16, 1;

M. Griffiths, 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.

Cf. A241500, A291117, A291118, A291119.

Formula including proof is at web link.

cp's OEIS Frontend

A035098 Near-Bell numbers: partitions of an n-multiset with multiplicities 1, 1, 1, ..., 1, 2.

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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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A291118 Triangle read by rows: T(n,k) number of ways of partitioning the (n+3)-element multiset {1,1,1,1,2,3,...,n} into exactly k nonempty parts, n >= 0 and 1 <= k <= n + 3.

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A291119 Triangle read by rows: T(n,k) number of ways of partitioning the (n+4)-element multiset {1,1,1,1,1,2,3,...,n} into exactly k nonempty parts, n >= 0 and 1 <= k <= n + 4.

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A291120 Triangle read by rows: T(n,k) number of ways of partitioning the (n+5)-element multiset {1,1,1,1,1,1,2,3,...,n} into exactly k nonempty parts, n >= 0 and 1 <= k <= n + 5.

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