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A247491 Crossing partitions of {1,2,...,n} that contain no singletons. 3
0, 0, 0, 0, 1, 5, 26, 126, 624, 3193, 17119, 96668, 576104, 3621982, 23980620, 166805068, 1215842905, 9263445775, 73599067250, 608471202527, 5224252803246, 46499854580107, 428369819029085, 4078345518655015, 40073659206668916, 405885206895408576, 4232705116291188276 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

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.

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..163

Peter Luschny, Set partitions

FORMULA

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

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

EXAMPLE

The crossing partitions of {1,2,3,4,5} that contain no singletons are: [13|245], [14|235], [24,135], [25|134], [35|124].

MAPLE

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);

MATHEMATICA

Table[Sum[(-1)^(n-k)*Binomial[n, k]*(BellB[k]-CatalanNumber[k]), {k, 0, n}], {n, 0, 26}] (* Indranil Ghosh, Mar 04 2017 *)

PROG

(Sage)

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

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

(PARI)

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

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

CROSSREFS

Cf. A016098, A005043, A000110, A000296.

Sequence in context: A254825 A272123 A285905 * A244617 A003583 A033115

Adjacent sequences:  A247488 A247489 A247490 * A247492 A247493 A247494

KEYWORD

nonn

AUTHOR

Peter Luschny, Sep 25 2014

STATUS

approved

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Last modified May 27 21:49 EDT 2020. Contains 334671 sequences. (Running on oeis4.)