A035098 - OEIS

%I #48 Aug 06 2021 15:03:10

%S 1,2,4,11,36,135,566,2610,13082,70631,407846,2504071,16268302,

%T 111378678,800751152,6027000007,47363985248,387710909055,

%U 3298841940510,29119488623294,266213358298590,2516654856419723,24566795704844210

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

%C 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.

%C 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.

%C Intermediate sequences are the number of ways of partitioning an n-multiset with multiplicities some partition of n.

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

%H Alois P. Heinz, <a href="/A035098/b035098.txt">Table of n, a(n) for n = 1..576</a> (first 200 terms from Vincenzo Librandi)

%H M. Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Griffiths/griffiths7.html">Generalized Near-Bell Numbers</a>, JIS 12 (2009) 09.5.7.

%H M. Griffiths, I. Mezo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Griffiths/griffiths11.html">A generalization of Stirling Numbers of the Second Kind via a special multiset</a>, JIS 13 (2010) #10.2.5.

%H Martin Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Griffiths/griffiths31.html">Generating Functions for Extended Stirling Numbers of the First Kind</a>, Journal of Integer Sequences, 17 (2014), #14.6.4.

%F 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]

%F a(n) ~ Bell(n)/2 * (1 + LambertW(n)/n). - _Vaclav Kotesovec_, Jul 28 2021

%e 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}}.

%p 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

%t f[n_] := Sum[ StirlingS2[n, k] ((k + 1) (k + 2)/2 + 1), {k, 0, n}]; Array[f, 22, 0]

%t f[n_] := (BellB[n] + BellB[n + 1] + BellB[n + 2])/2; Array[f, 22, 0]

%t 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 *)

%t Join[{1},Total[#]/2&/@Partition[BellB[Range[0,30]],3,1]] (* _Harvey P. Dale_, Jan 02 2019 *)

%Y Cf. A000070, A000110, A059606, A087648.

%Y Row sums of A241500.

%Y Column 1 of array in A322765.

%Y Row n=2 of A346426.

%K nonn

%O 1,2

%A _George Beck_

%E More terms from _Vladeta Jovovic_, Sep 23 2003