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A024167
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n!*(1 - 1/2 + 1/3 - .. + c/n), where c = (-1)^(n+1).
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17
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1, 1, 5, 14, 94, 444, 3828, 25584, 270576, 2342880, 29400480, 312888960, 4546558080, 57424792320, 948550176000, 13869128448000, 256697973504000, 4264876094976000, 87435019510272000, 1627055289796608000
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OFFSET
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1,3
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COMMENTS
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Stirling transform of (-1)^n*a(n-1)=[0,1,-1,5,-14,94,...] is A000629(n-2)=[0,1,2,6,26,...]. - Michael Somos, Mar 04 2004
Stirling transform of a(n)=[1,1,5,14,94,...] is A052882(n)=[1,2,9,52,375,...]. - Michael Somos, Mar 04 2004
a(n) is the number of n-permutations that have a cycle with length greater than n/2. [Geoffrey Critzer, May 28 2009]
From Jens Voß, May 07 2010: (Start)
A024167(4n) is divisible by 6n+1 for all n >= 1; the quotient of A024167(4n) and 6n+1 is A177188(n).
A024167(4n+3) is divisible by 6n+5 for all n >= 0; the quotient of A024167(4n+3) and 6n+5 is A177174(n). (End)
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LINKS
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Table of n, a(n) for n=1..20.
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FORMULA
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E.g.f.: ln(1+x)/(1-x). - Vladeta Jovovic, Aug 25 2002
a(n)=a(n-1)+a(n-2)*(n-1)^2, n>1. - Michael Somos, Oct 29, 2002
b(n) = n! satisfies the above recurrence with b(1) = 1, b(2) = 2. This gives the finite continued fraction expansion a(n)/n! = 1/(1+1^2/(1+2^2/(1+3^2/(1+...+(n-1)^2/1)))). Cf. A142979. - Peter Bala, Jul 17 2008
a(n) = A081358(n) - A092691(n). [Gary Detlefs, Jul 09 2010]
E.g.f. x/(x-1)/G(0) where G(k)= -1 + (x-1)*k + x*(k+1)^2/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Aug 18 2012
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MATHEMATICA
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f[k_] := k (-1)^(k + 1)
t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 18}] (* A024167 signed *)
(* Clark Kimberling, Dec 30 2011 *)
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PROG
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(PARI) a(n)=if(n<0, 0, n!*polcoeff(log(1+x+x*O(x^n))/(1-x), n))
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CROSSREFS
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Cf. A000254.
Cf. A142979.
Cf. A177174, A177188. [From Jens Voß, May 07 2010]
Sequence in context: A197797 A224245 A183307 * A077262 A184439 A058072
Adjacent sequences: A024164 A024165 A024166 * A024168 A024169 A024170
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KEYWORD
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nonn,easy,changed
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AUTHOR
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Clark Kimberling
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EXTENSIONS
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More terms from Benoit Cloitre, Jan 27 2002
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STATUS
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approved
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