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A247494
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Number of crossing partitions of {1,2,...,n} that contain singletons.
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3
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0, 0, 0, 0, 0, 5, 45, 322, 2086, 13092, 82060, 523116, 3429481, 23279555, 164244262, 1206458632, 9228941572, 73471779239, 608000100209, 5222503739340, 46493341311706, 428345495309624, 4078254436854598, 40073317276815681, 405883920183989049, 4232700263388189325
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OFFSET
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0,6
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COMMENTS
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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.
Also number of crossing partitions of {1,2,...,n} that contain cyclical adjacencies. a(5) = 5, [124|35, 134|25, 135|24, 13|245, 14|235]. - Yuchun Ji, Nov 13 2020
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LINKS
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FORMULA
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a(n) = Sum_{k = 0..n-1} (-1)^(n-k+1)*binomial(n,k)*(Bell(k)-Catalan(k)).
a(n) = A000296(n+1) - A106640(n-1), for n>0 (i.e., remove the non-crossing partitions from the cyclical adjacencies partitions). - Yuchun Ji, Nov 11 2020
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EXAMPLE
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The crossing partitions of {1,2,3,4,5} that contain singletons are: [1|24|35], [2|14|35], [3|14|25], [4|13|25], [5|13|24].
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MAPLE
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A247494 := n -> add((-1)^(n-k+1)*combinat:-bell(k+1), k=0..n-1) + (-1)^n*hypergeom([-n, 1/2], [2], 4) - binomial(2*n, n)/(n+1):
seq(round(evalf(A247494(n), 100)), n=0..25);
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MATHEMATICA
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Table[Sum[(-1)^(n-k+1)*Binomial[n, k]*(BellB[k]-CatalanNumber[k]), {k, 0, n-1}], {n, 0, 25}] (* Indranil Ghosh, Mar 04 2017 *)
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PROG
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(Sage)
A247494 = lambda n: sum((-1)^(n-k+1)*binomial(n, k)*(bell_number(k)-catalan_number(k)) for k in (0..n-1))
(PARI)
B(n) = sum(k=0, n, stirling(n, k, 2));
a(n) = sum(k=0, n-1, (-1)^(n-k+1)*binomial(n, k)*(B(k) - binomial(2*k, k)/(k+1))); \\ Indranil Ghosh, Mar 04 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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