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A130915
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Number of permutations in the symmetric group S_n in which cycle lengths are odd and greater than 1.
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6
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1, 0, 0, 2, 0, 24, 40, 720, 2688, 42560, 245376, 4072320, 31672960, 569935872, 5576263680, 109492807424, 1290701905920, 27616577064960, 380852962029568, 8845627365089280, 139696582370328576, 3506062524305162240, 62387728088875499520, 1684340707284076756992
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OFFSET
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0,4
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LINKS
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FORMULA
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E.g.f.: exp(-x)*sqrt((1+x)/(1-x)).
a(n) = (-1)^n*Sum_{k = 0..n} (1 if n = k, otherwise (-1)^(n + k)*(n - k)!*Sum_{i = 1..n - k} Sum_{j = i..n - k} 2^(j - i)*Stirling1(j, i)*binomial(n - k - 1, j - 1)/j!*binomial(n, k)). - Detlef Meya, Jan 18 2024
a(n) = (n-1)*(n-2)*(a(n-2)+a(n-3)) for n>=3. - Alois P. Heinz, Jan 18 2024
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EXAMPLE
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a(3)=2 because we have (123) and (132).
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MAPLE
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g:=exp(-x)*sqrt((1+x)/(1-x)): gser:=series(g, x=0, 30): seq(factorial(n)*coeff(gser, x, n), n=0..20); # Emeric Deutsch, Aug 25 2007
# second Maple program:
a:= proc(n) option remember;
`if`(n<3, 1/2, a(n-2)+a(n-3))*(n-1)*(n-2)
end:
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MATHEMATICA
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nn=20; Drop[Range[0, nn]!CoefficientList[Series[((1+x)/(1-x))^(1/2)Exp[-x], {x, 0, nn}], x], 1] (* Geoffrey Critzer, Dec 15 2012 *)
a[n_] := (-1)^n*Sum[If[n==k, 1, (-1)^(n + k)*(n - k)!*Sum[Sum[2^(j - i)*StirlingS1[j, i]*Binomial[n - k - 1, j - 1]/j!, {j, i, n - k}], {i, 1, n - k}]*Binomial[n, k]], {k, 0, n}]; Flatten[Table[a[n], {n, 1, 20}]] (* Detlef Meya, Jan 18 2024 *)
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PROG
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(PARI) my(x='x+O('x^33)); Vec(serlaplace(exp(-x)*sqrt((1+x)/(1-x)))) \\ Joerg Arndt, Jan 18 2024
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CROSSREFS
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KEYWORD
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easy,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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