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A081358
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Expansion of log((1+x)/(1-x))/(2*(1-x)).
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9
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0, 1, 2, 8, 32, 184, 1104, 8448, 67584, 648576, 6485760, 74972160, 899665920, 12174658560, 170445219840, 2643856588800, 42301705420800, 740051782041600, 13320932076748800, 259500083163955200, 5190001663279104000
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OFFSET
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0,3
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COMMENTS
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Number of cycles of odd cardinality in all permutations of [n]. Example: a(3)=8 because among (1)(2)(3), (1)(23), (12)(3), (13)(2), (132), (123) we have eight cycles of odd length. - Emeric Deutsch, Aug 12 2004
a(n) is a function of the harmonic numbers. a(n)=n!*h(n)-n!/2 * h(floor(n/2)), where h(n) = sum(1/k,k=1..n) [From Gary Detlefs, Aug 06 2010]
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REFERENCES
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I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, Exercise 3.3.13
B. A. Kuperschmidt, ... And free lunch for all. A review of Bruce C. Berndt's Ramanujan's notebooks, J. Nonlinear Math. Phys., 7 (2000), R7-R37. MR 2002d:33024.
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 0..30
B. A. Kuperschmidt, ... And free lunch for all.
B. A. Kuperschmidt, Journal of Non linear Mathematical Physics 2000 v.7 no.2, A Review of Bruce C.Berndt's Ramanujan's Notebooks parts I-V
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FORMULA
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E.g.f.: log((1+x)/(1-x))/(2(1-x)).
a(n) = n! * sum[ k=0..n, k odd ] 1/k.
a(n) = n!/2*(Psi(ceil(n/2)+1/2)+gamma+2*ln(2)). - Vladeta Jovovic, Oct 20 2003
a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*2^(k-1)*binomial(n, k)/k. - Vladeta Jovovic, Aug 12 2005
a(n) = n*a(n-1) + ((-1)^(n+1)+1)/2*(n-1)! [From Gary Detlefs, Aug 06 2010]
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MATHEMATICA
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nn = 20; Range[0, nn]! CoefficientList[
D[Series[(1 - x^2)^(-1/2) ((1 + x)/(1 - x))^(y/2), {x, 0, nn}], y] /. y -> 1, x] (*Geoffrey Critzer, Aug 27 2012*)
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PROG
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(PARI) a(n)=if(n<1, 0, n!*polcoeff(log(1+2/(-1+1/(x+x*O(x^n))))/(1-x)/2, n))
(PARI) {a(n)=if(n<0, 0, n!*sum(k=1, n, (k%2)/k))} /* Michael Somos Sep 19 2006 */
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CROSSREFS
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A049034(n)=a(2n+1). Cf. A151884, A092691.
a(n) = A000254(n) - A092691(n) [From Gary Detlefs, Aug 06 2010]
Sequence in context: A081561 A009753 A141202 * A206303 A048855 A062797
Adjacent sequences: A081355 A081356 A081357 * A081359 A081360 A081361
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KEYWORD
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nonn,changed
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AUTHOR
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Michael Somos, Mar 18 2003
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STATUS
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approved
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