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A092043
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Numerator of n!/n^2.
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5
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1, 1, 2, 3, 24, 20, 720, 630, 4480, 36288, 3628800, 3326400, 479001600, 444787200, 5811886080, 81729648000, 20922789888000, 19760412672000, 6402373705728000, 6082255020441600, 115852476579840000, 2322315553259520000
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OFFSET
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1,3
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COMMENTS
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Numerator of expansion of dilog(x) = Li_2(x) = - int(0, x, log(1-t)/t*dt). See the Weisstein link.
E.g.f. of {a(n)/A014973(n)}_{n>=1} is Li_2(x) (with 0 for n=0).
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..200
A. N. Kirillov, Dilogarithm identities, arXiv:hep-th/9408113, 1994.
Eric Weisstein's World of Mathematics, Dilogarithm
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FORMULA
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From Wolfdieter Lang, Apr 28 2017: (Start)
a(n) = numerator(n!/n^2) = numerator((n-1)!/n), n >= 1. See the name.
E.g.f. {a(n)/A014973(n)}_{n>=1} with 0 for n=0 is Li_2(x). See the comment.
(-1)^n*a(n+1)/A014973(n+1) = (-1)^n*n!/(n+1) = Sum_{k=0..n} Stirling1(n, k)*Bernoulli(k), with Stirling1 = A048994 and Bernoulli(k) = A027641(k)/A027642(k), n >= 0. From inverting the formula for B(k) in terms of Stirling2 = A048993.(End)
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MATHEMATICA
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Table[Numerator[n!/n^2], {n, 1, 40}] (* Vincenzo Librandi, Apr 15 2014 *)
Table[(n-1)!/n, {n, 30}]//Numerator (* Harvey P. Dale, Apr 03 2018 *)
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PROG
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(PARI) a(n)=numerator(n!/n^2)
(PARI) a(n)=numerator(polcoeff(serlaplace(dilog(x)), n))
(MAGMA) [Numerator(Factorial(n)/n^2): n in [1..30]]; // Vincenzo Librandi, Apr 15 2014
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CROSSREFS
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Denominator is in A014973. Cf. A001819.
Sequence in context: A170909 A160606 A099617 * A055067 A037319 A032811
Adjacent sequences: A092040 A092041 A092042 * A092044 A092045 A092046
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KEYWORD
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nonn,easy,frac
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AUTHOR
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Ralf Stephan, Mar 28 2004
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EXTENSIONS
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Comment rewritten by Wolfdieter Lang, Apr 28 2017
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STATUS
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approved
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