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A248682
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Decimal expansion of Sum_{n >= 0} (floor(n/2)!)^2/n!.
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9
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2, 9, 4, 5, 5, 9, 9, 4, 3, 4, 8, 7, 4, 8, 6, 0, 3, 1, 1, 6, 3, 9, 1, 8, 0, 6, 7, 3, 4, 5, 9, 6, 9, 3, 9, 8, 4, 2, 5, 2, 5, 0, 3, 3, 3, 1, 6, 3, 7, 9, 9, 1, 6, 2, 2, 7, 2, 8, 7, 8, 6, 6, 0, 9, 2, 3, 3, 8, 8, 7, 2, 7, 2, 1, 1, 2, 3, 1, 4, 5, 6, 3, 2, 7, 4, 7
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OFFSET
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1,1
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COMMENTS
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Limit_{x -> inf} Sum {n=0..inf} (Floor[n/x])!^x/n! = e (A001113).
Let n} denote the swinging factorial A056040(n), then the constant equals Sum_{n>=0} 1/n} and is sometimes called the swinging constant e}. ("e}" is written in TeX $e\wr$). For a proof that it equals 3^(1/2)*(2/3)^3*Pi + 4/3 see the link to Mathematics Stack Exchange. - Peter Luschny, Jul 22 2022
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LINKS
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FORMULA
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Equals Sum_{n >= 0} (n!^2)*p(2,n)/(2*n + 1)!, where p(k,n) is defined at A248664.
Equals Sum_{n >= 0} (floor(n/2)!)^2/n! = Sum_(n >= 1) (3n^2 - 7n + 6)/C(2n, n) = 4/3 + 8*Pi/sqrt(243). - Robert G. Wilson v, Feb 11 2016
Equals 1 + Integral_{x>=0} 1/(x^2 - x + 1)^2 dx. - Amiram Eldar, Nov 16 2021
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EXAMPLE
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2.94559943487486031163918067345969398425250...
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MATHEMATICA
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RealDigits[Sum[(Floor[n/2])!^2/n!, {n, 0, 400}], 10, 111][[1]]
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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