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A247491 Number of crossing partitions of {1,2,...,n} that contain no singletons.

%I A247491 #37 Dec 18 2020 14:12:02
%S A247491 0,0,0,0,1,5,26,126,624,3193,17119,96668,576104,3621982,23980620,
%T A247491 166805068,1215842905,9263445775,73599067250,608471202527,
%U A247491 5224252803246,46499854580107,428369819029085,4078345518655015,40073659206668916,405885206895408576,4232705116291188276
%N A247491 Number of crossing partitions of {1,2,...,n} that contain no singletons.
%C A247491 A partition p of the set {1,2,...,n} whose elements are arranged in their natural order, is crossing if there exist four numbers 1 <= i < k < j < l <= n such that i and j are in the same block, k and l are in the same block, but i,j and k,l belong to two different blocks.
%C A247491 Also the number of crossing partitions of {1,2,...,n} that contain no cyclical adjacencies. e.g., a(5) = 5, [13|24|5, 13|25|4, 14|25|3, 14|2|35, 1|24|35]. - _Yuchun Ji_, Nov 13 2020
%H A247491 Indranil Ghosh, <a href="/A247491/b247491.txt">Table of n, a(n) for n = 0..163</a>
%H A247491 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/SetPartitions">Set partitions</a>
%F A247491 a(n) = Sum_{k=0..n} (-1)^(n-k)*C(n,k)*(Bell(k)-Catalan(k)).
%F A247491 a(n) = A000296(n) - A005043(n).
%F A247491 a(n) = A016098(n) - A247494(n); i.e., remove the partitions with cyclical adjacencies from the crossing partitions. - _Yuchun Ji_, Nov 17 2020
%e A247491 The crossing partitions of {1,2,3,4,5} that contain no singletons are: [13|245], [14|235], [24,135], [25|134], [35|124].
%p A247491 A247491 := n -> (-1)^n-add((-1)^(n-k)*combinat:-bell(k), k = 0..n-1) - (-1)^n*hypergeom([-n, 1/2], [2], 4); seq(round(evalf(A247491(n), 100)), n=0..27);
%t A247491 Table[Sum[(-1)^(n-k)*Binomial[n,k]*(BellB[k]-CatalanNumber[k]), {k,0,n}], {n, 0, 26}] (* _Indranil Ghosh_, Mar 04 2017 *)
%o A247491 (Sage)
%o A247491 A247491 = lambda n: sum((-1)^(n-k)*binomial(n,k)*(bell_number(k) - catalan_number(k)) for k in (0..n))
%o A247491 [A247491(n) for n in range(27)]
%o A247491 (PARI)
%o A247491 B(n) = sum(k=0, n, stirling(n,k,2));
%o A247491 a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n,k)*(B(k)-binomial(2*k,k)/(k+1))); \\ _Indranil Ghosh_, Mar 04 2017
%Y A247491 Cf. A016098, A005043, A000110, A000296, A247494.
%K A247491 nonn
%O A247491 0,6
%A A247491 _Peter Luschny_, Sep 25 2014