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A186760
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Number of cycles that are either nonincreasing or of length 1 in all permutations of {1,2,...,n}. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)<b(2)<b(3)<... .
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7
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0, 1, 2, 7, 33, 188, 1247, 9448, 80623, 765926, 8022139, 91872328, 1142384735, 15330003154, 220847064955, 3399884265524, 55705822616383, 967921774366510, 17778279366693179, 344189681672898400, 7005438733866799999, 149547115419379439978, 3341127481398057119515
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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E.g.f.: (1+z-exp(z)-log(1-z))/(1-z).
a(n) ~ n! * (log(n) + gamma + 2 - exp(1)), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 08 2013
D-finite with recurrence a(n) +(-2*n-1)*a(n-1) +(n^2+2*n-4)*a(n-2) +(-2*n^2+6*n-3)*a(n-3) +((n-3)^2)*a(n-4)=0. - R. J. Mathar, Jul 26 2022
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EXAMPLE
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a(3) = 7 because in (1)(2)(3), (1)(23), (12)(3), (13)(2), (123), and (132) we have a total of 3 + 1 + 1 + 1 + 0 + 1 = 7 cycles that are either of length 1 or nonincreasing.
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MAPLE
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g := (1+z-exp(z)-ln(1-z))/(1-z): gser := series(g, z = 0, 25): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 22);
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MATHEMATICA
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CoefficientList[Series[(1+x-E^x-Log[1-x])/(1-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 08 2013 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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