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A245141
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Number of endofunctions f on [2n] that are self-inverse on [n].
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3
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1, 3, 50, 1626, 83736, 6026120, 571350096, 67996818960, 9862902275456, 1700092943088768, 342087177215788800, 79115601821198404352, 20779757607847901690880, 6133520505473954148381696, 2017134796016735182500521984, 733523863838078950241395968000
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OFFSET
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0,2
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COMMENTS
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a(n) counts endofunctions f:{1,...,2n}-> {1,...,2n} with f(f(i))=i for all i in {1,...,n}.
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LINKS
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FORMULA
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a(n) = Sum_{i=0..n} C(n,i)^2 * i! * A000085(n-i) * (2*n)^(n-i).
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EXAMPLE
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a(1) = 3: (1,1), (1,2), (2,1).
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MAPLE
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g:= proc(n) g(n):= `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
a:= n-> add(binomial(n, i)^2*i!*g(n-i)*(2*n)^(n-i), i=0..n):
seq(a(n), n=0..20);
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MATHEMATICA
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Join[{1}, Table[n! * Sum[Binomial[n, k] * 2^k * n^k* Sum[1/((k - 2*j)!*2^j*j!), {j, 0, Floor[k/2]}], {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Dec 05 2021 *)
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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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