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

Author

George Beck

Keywords

• nonn

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.

Examples

			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}}.
		

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.

Crossrefs

Cf. A000070, A000110, A059606, A087648.

Row sums of A241500.

Column 1 of array in A322765.

Row n=2 of A346426.

Programs

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 *)

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

Extensions

More terms from Vladeta Jovovic, Sep 23 2003