cp's OEIS Frontend

A086365 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}.

%I A086365 #22 Sep 02 2025 21:54:36
%S A086365 1,4,15,75,428,2781,20093,159340,1372163,12725447,126238060,
%T A086365 1332071241,14881206473,175297058228,2169832010759,28136696433171,
%U A086365 381199970284620,5383103100853189,79065882217154085,1205566492711167004,19049651311462785947
%N A086365 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 A086365 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 A086365 E.g.f. (for offset 1): -1 + exp(-x + Sum_{j=1..2} (exp(j*x)-1)/j). - _Joerg Arndt_, Apr 29 2011
%e A086365 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 A086365 (PARI)
%o A086365 x = 'x + O('x^16);
%o A086365 egf = -1 + exp(-x+sum(j=1,2,(exp(j*x)-1)/j))
%o A086365 /* egf == +x +2*x^2 +5/2*x^3 +25/8*x^4 +... (i.e., for offset 1) */
%o A086365 Vec( serlaplace(egf) )
%o A086365 /* _Joerg Arndt_, Apr 29 2011 */
%Y A086365 Cf. A002872, A086364.
%K A086365 easy,nonn,changed
%O A086365 0,2
%A A086365 _James East_, Sep 04 2003
%E A086365 More terms from _Joerg Arndt_, Apr 29 2011
%E A086365 Definition shortened by _M. F. Hasler_, Oct 21 2012