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

1,2

Comments

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.

References

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

Links

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.

Formula

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

Example

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}}.

Maple

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

Mathematica

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 *)

Crossrefs

Cf. A000070, A000110, A059606, A087648.

Row sums of A241500.

Column 1 of array in A322765.

Row n=2 of A346426.

Sequence in context: A193058 A179379 A086611 * A328425 A174107 A253927

Adjacent sequences: A035095 A035096 A035097 * A035099 A035100 A035101

Keyword

nonn

Author

George Beck

Extensions

More terms from Vladeta Jovovic, Sep 23 2003

Status

approved