OFFSET
0,2
COMMENTS
Number of partitions of n+4 with at least one singleton and with the smallest element in a singleton equal to 4. Alternatively, number of partitions of n+4 with at least one singleton and with the largest element in a singleton equal to n+1. - Olivier GERARD, Oct 29 2007
REFERENCES
Olivier GĂ©rard and Karol A. Penson, A budget of set partition statistics, in preparation, unpublished as of Sep 22 2011.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..215
Cohn, Martin; Even, Shimon; Menger, Karl, Jr.; Hooper, Philip K.; On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly 69 (1962), no. 8, 782--785. MR1531841.
Cohn, Martin; Even, Shimon; Menger, Karl, Jr.; Hooper, Philip K.; On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly 69 (1962), no. 8, 782--785. MR1531841. [Annotated scanned copy]
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.
FORMULA
G.f.: -(1-x+x^2)/x^2 + (1-x)^3/x^2/(G(0)-x) where G(k) = 1 - x*(k+1)/(1 - x/G(k+1) ); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 26 2013
From Vaclav Kotesovec, Jul 28 2021: (Start)
a(n) = Bell(n+3) - 3*Bell(n+2) + 3*Bell(n+1) - Bell(n).
a(n) ~ n^3 * Bell(n) / LambertW(n)^3 * (1 - 3*LambertW(n)/n). (End)
MAPLE
a:= n-> add ((-1)^(k+1) *binomial(3, k) *combinat['bell'](n+k), k=0..3): seq (a(n), n=0..20); # Alois P. Heinz, Sep 05 2008
MATHEMATICA
Differences[BellB[Range[0, 30]], 3] (* Harvey P. Dale, Apr 21 2011 *)
PROG
(Python)
# requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
from itertools import accumulate
A011966_list, blist, b = [1], [2, 3, 5], 5
for _ in range(1000):
....blist = list(accumulate([b]+blist))
....b = blist[-1]
....A011966_list.append(blist[-4]) # Chai Wah Wu, Sep 20 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved