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A140456
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a(n) is the number of indecomposable involutions of length n.
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6
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1, 1, 1, 3, 7, 23, 71, 255, 911, 3535, 13903, 57663, 243871, 1072031, 4812575, 22278399, 105300287, 510764095, 2527547455, 12794891007, 66012404863, 347599231103, 1863520447103, 10178746224639, 56548686860543, 319628408814847, 1835814213846271
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OFFSET
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1,4
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COMMENTS
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An involution is a self-inverse permutation. A permutation of [n] = {1, 2, ..., n} is indecomposable if it does not fix [j] for any 0 < j < n.
G.f. of a(n+1) is 1/(1-x-2x^2/(1-x-3x^2/(1-x-4x^2/(1-x-5x^2/(1-...))))) (continued fraction).
a(n+1) is the binomial transform of the aeration of A000698(n+1). Hankel transform of a(n+1) is A000178(n+1). (End)
a(n) is the INVERTi transform of A000085(n+1)
a(n) is also the moment of order n for the density: sqrt(2/Pi^3)*exp((x-1)^2/2)/(1-(erf(I*(x-1)/sqrt(2)))^2).
More generally, if c(n)=int(x^n*rho(x),x=a..b) with rho(x) a probability density function of class C1, then the INVERTi transform of (c(1),..c(n),..) starting at n=2 gives the moments of mu(x) = rho(x) / ((s(x))^2+(Pi*rho(x))^2) with s(x) = int( rho'(t)*log(abs(1-t/x)), t=a..b) + rho(b)*log(x/(b-x)) + rho(a)*log((x-a)/x).
(End)
For n>1 sum over all Motzkin paths of length n-2 of products over all peaks p of (x_p+y_p)/y_p, where x_p and y_p are the coordinates of peak p. - Alois P. Heinz, May 24 2015
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LINKS
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FORMULA
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G.f.: 1 - 1/I(x), where I(x) is the ordinary generating function for involutions (A000085).
G.f.: Q(0) +1/x, where Q(k) = 1 - 1/x - (k+1)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Sep 16 2013
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EXAMPLE
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The unique indecomposable involution of length 3 is 321. The indecomposable involutions of length 4 are 3412, 4231 and 4321.
G.f. = x + x^2 + 3*x^3 + 7*x^4 + 23*x^5 + 71*x^6 + 255*x^7 + 911*x^8 + ...
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MAPLE
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b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
`if`(x=0, 1, b(x-1, y-1, false)*`if`(t, (x+y)/y, 1)
+ b(x-1, y, false) + b(x-1, y+1, true)))
end:
a:= n-> `if`(n=1, 1, b(n-2, 0, false)):
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MATHEMATICA
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CoefficientList[Series[1 - 1/Total[CoefficientList[Series[E^(x + x^2/2), {x, 0, 50}], x] * Range[0, 50]! * x^Range[0, 50]], {x, 0, 50}], x]
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CROSSREFS
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Cf. A000085 (involutions), A000698 (indecomposable fixed-point free involutions), and A003319 (indecomposable permutations).
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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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