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n-th Bell number of type D: Number of symmetric partitions of {-n,...,n}\{0} such that none of the subsets is of the form {j,-j}.
2

%I #18 Jan 08 2018 01:47:44

%S 1,4,15,75,428,2781,20093,159340,1372163,12725447,126238060,

%T 1332071241,14881206473,175297058228,2169832010759,28136696433171,

%U 381199970284620,5383103100853189,79065882217154085,1205566492711167004,19049651311462785947

%N n-th Bell number of type D: Number of symmetric partitions of {-n,...,n}\{0} such that none of the subsets is of the form {j,-j}.

%C A partition of {-n,...,-1,1,...,n} into nonempty subsets X_1,...,X_r is called `symmetric' if for each i -X_i = X_j for some j. a(n) is the number of such symmetric partitions such that none of the X_i are of the form {j,-j}.

%F E.g.f. (for offset 1): -1 + exp(-x+sum(j=1,2,(exp(j*x)-1)/j)) [_Joerg Arndt_, Apr 29 2011]

%e a(2)=4 because the relevant partitions of {-2,-1,1,2} are {-2|-1|1|2}, {-2,-1|1,2}, {-2,1|-1,2} and {-2,-1,1,2}.

%o (PARI)

%o x = 'x + O('x^16);

%o egf = -1 + exp(-x+sum(j=1,2,(exp(j*x)-1)/j))

%o /* egf == +x +2*x^2 +5/2*x^3 +25/8*x^4 +... (i.e., for offset 1) */

%o Vec( serlaplace(egf) )

%o /* _Joerg Arndt_, Apr 29 2011 */

%Y Cf. A002872, A086364.

%K easy,nonn

%O 0,2

%A _James East_, Sep 04 2003

%E More terms from Joerg Arndt, Apr 29 2011.

%E Definition shortened by _M. F. Hasler_, Oct 21 2012