A247491 - OEIS

%I #37 Dec 18 2020 14:12:02

%S 0,0,0,0,1,5,26,126,624,3193,17119,96668,576104,3621982,23980620,

%T 166805068,1215842905,9263445775,73599067250,608471202527,

%U 5224252803246,46499854580107,428369819029085,4078345518655015,40073659206668916,405885206895408576,4232705116291188276

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

%C 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 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 Indranil Ghosh, <a href="/A247491/b247491.txt">Table of n, a(n) for n = 0..163</a>

%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/SetPartitions">Set partitions</a>

%F a(n) = Sum_{k=0..n} (-1)^(n-k)*C(n,k)*(Bell(k)-Catalan(k)).

%F a(n) = A000296(n) - A005043(n).

%F a(n) = A016098(n) - A247494(n); i.e., remove the partitions with cyclical adjacencies from the crossing partitions. - _Yuchun Ji_, Nov 17 2020

%e 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 := 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 Table[Sum[(-1)^(n-k)*Binomial[n,k]*(BellB[k]-CatalanNumber[k]), {k,0,n}], {n, 0, 26}] (* _Indranil Ghosh_, Mar 04 2017 *)

%o (Sage)

%o A247491 = lambda n: sum((-1)^(n-k)*binomial(n,k)*(bell_number(k) - catalan_number(k)) for k in (0..n))

%o [A247491(n) for n in range(27)]

%o (PARI)

%o B(n) = sum(k=0, n, stirling(n,k,2));

%o 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 Cf. A016098, A005043, A000110, A000296, A247494.

%K nonn

%O 0,6

%A _Peter Luschny_, Sep 25 2014