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A188144
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Binomial transform A140456(n+1) (indecomposable involutions).
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1
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1, 2, 6, 20, 74, 292, 1218, 5308, 24034, 112484, 542346, 2686268, 13639226, 70863652, 376208706, 2038335580, 11259474754, 63353211332, 362819139978, 2113410084188, 12513610048154, 75274067489284, 459782361574146, 2850369932150908, 17926893505949986, 114337654086861092
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OFFSET
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0,2
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COMMENTS
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a(n) is also the INVERTi transform of A005425(n+1) (self-inverse partial permutations) starting at n=2.
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LINKS
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FORMULA
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a(n) is the moment of order n for the probability density function: sqrt(2/Pi^3)*exp((x-2)^2/2)/(1+(erfi((x-2)/sqrt(2)))^2) over the interval -infinity..infinity, with erfi the imaginary error function.
G.f.: A(x) = (1 - 2*x - G(0))/x^2; G(k) = 1 - 2*x - x^2*(k+1)/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jan 26 2012
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MAPLE
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b:= proc(n) b(n):= `if`(n<2, n+1, 2*b(n-1) + (n-1)*b(n-2)) end:
g:= proc(n) g(n):= `if`(n<1, -1, -add(g(n-i)*b(i), i=1..n)) end:
a:= n-> g(n+2):
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MATHEMATICA
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nmax = 18; A140456 = CoefficientList[ Series[1 - 1/Total[ CoefficientList[ Series[Exp[x^2/2 + x], {x, 0, nmax + 2}], x]*Range[0, nmax + 2]!* x^Range[0, nmax + 2]], {x, 0, nmax + 2}], x]; a[n_] := Sum[ Binomial[n, k]*A140456[[k + 3]], {k, 0, n}]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Jul 03 2013 *)
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PROG
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(PARI) seq(n)={Vec(1 - 2*x - 1/serlaplace(exp( 2*x + x^2/2 + O(x^3*x^n) )))} \\ Andrew Howroyd, Jan 06 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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