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A279055
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Self-convolution of squares of factorial numbers (A001044).
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1
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1, 2, 9, 80, 1240, 30240, 1071504, 51996672, 3307723776, 266872320000, 26615381760000, 3214252921651200, 462189467175321600, 78024380924038348800, 15279632043682406400000, 3435553774431004262400000, 879010223384483132866560000, 253916900613208108255150080000
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OFFSET
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0,2
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COMMENTS
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a(n) = (n!)^2 * Sum_{i=0..n} (binomial(n,i)^(-2)).
Consider a triangle ABC with area p. Let points X, Y, Z be randomly and uniformly chosen on sides BC, CA, BA. Let r = area of XYZ. Then the average or expected value of (r/p)^n = a(n)/(n!^2 * (n+1)^3).
a(n) = (3*(n+1)^4 *(n!)^4 /(2n+3)!) * Sum_{i=1..n+1} ((1/i)* binomial(2i, i)), see Sprugnoli Formula 5.2 as noted by Markus Scheuer.
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LINKS
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FORMULA
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a(n) = Sum_{i=0..n} (i! * (n-i)!)^2.
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MATHEMATICA
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Table[Sum[(k!*(n-k)!)^2, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 05 2016 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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