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A011966 Third differences of Bell numbers. 5
1, 5, 20, 87, 409, 2066, 11155, 64077, 389946, 2504665, 16923381, 119928232, 888980293, 6876320041, 55382419676, 463539664643, 4024626253845, 36189297168874, 336513491259647, 3231446022478129, 32004743929977258, 326548129128737469, 3428663026172389201 (list; graph; refs; listen; history; text; internal format)
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Olivier GĂ©rard and Karol A. Penson, A budget of set partition statistics, in preparation, unpublished as of Sep 22 2011.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

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

Cf. A000110, A005493. A106436, A011965.

Sequence in context: A026661 A099014 A219672 * A271096 A269716 A192249

Adjacent sequences:  A011963 A011964 A011965 * A011967 A011968 A011969

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified November 21 11:54 EST 2018. Contains 317447 sequences. (Running on oeis4.)