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A105748
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Number of ways to use the elements of {1,..,k}, 0<=k<=2n, once each to form a collection of n (possibly empty) sets, each with at most 2 elements.
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4
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1, 3, 10, 47, 313, 2744, 29751, 383273, 5713110, 96673861, 1830257967, 38326484944, 879473289521, 21944639630923, 591545277653354, 17131028946645255, 530424623323416617, 17485652721425863464, 611431929749388274471, 22604399407882099928577
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = Sum_{0<=i<=k<=n} (k+i)!/i!/(k-i)!/2^i.
G.f.: 1/U(0) where U(k)= 1 - 3*x + x^2 - x*4*k - x^2*(2*k+1)*(2*k+2)/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 06 2012
G.f.: 1/(1-x)/Q(0), where Q(k)= 1 - x - x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 19 2013
a(n) = 2*n*a(n-1) -(2*n-2)*a(n-2) -a(n-3) for n>2. - Alois P. Heinz, Mar 11 2015
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EXAMPLE
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a(2) = 10 = |{ {{},{}}, {{},{1}}, {{},{1,2}}, {{1},{2}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1,2},{3,4}}, {{1,3},{2,4}}, {{1,4},{2,3}} }|.
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MAPLE
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a:= proc(n) option remember; `if`(n<3, [1, 3, 10][n+1],
2*n*a(n-1)-(2*n-2)*a(n-2)-a(n-3))
end:
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MATHEMATICA
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Sum[(k+i)!/i!/(k-i)!/2^i, {k, 0, n}, {i, 0, k}]
(* Second program: *)
a[n_] := E*Sqrt[2/Pi]*Sum[BesselK[k + 1/2, 1], {k, 0, n}]; Table[a[n] // Round, {n, 0, 25}] (* Jean-François Alcover, Jul 15 2017 *)
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PROG
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(PARI) A105748(n) = sum(k=0, n, sum(i=0, k, binomial(k+i, k-i)*binomial(2*i, i)*i!>>i)) \\ M. F. Hasler, Oct 09 2012
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CROSSREFS
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Replacing "collection" by "sequence" gives A003011.
Replacing "sets" by "lists" gives A105747.
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KEYWORD
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nonn,easy
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AUTHOR
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Robert A. Proctor (www.math.unc.edu/Faculty/rap/), Apr 18 2005
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STATUS
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approved
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