VOOZH about

URL: https://oeis.org/A035098

⇱ A035098 - OEIS

login
A035098
Near-Bell numbers: partitions of an n-multiset with multiplicities 1, 1, 1, ..., 1, 2.
19
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.
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
Row sums of A241500.
Column 1 of array in A322765.
Row n=2 of A346426.
Sequence in context: A179379 A387692 A086611 * A328425 A174107 A253927
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Sep 23 2003
STATUS
approved