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A025167
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E.g.f: exp(x/(1-2*x))/(1-2*x).
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10
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1, 3, 17, 139, 1473, 19091, 291793, 5129307, 101817089, 2250495523, 54780588561, 1455367098923, 41888448785857, 1298019439099059, 43074477771208913, 1523746948247663611, 57229027745514785793, 2274027983943883110467
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OFFSET
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0,2
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COMMENTS
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Polynomials in A021009 evaluated at -2.
Also, a(n) is the number of signed permutations of length 2n that are equal to their reverse-complements and avoid the pattern (-2,-1). As a result, a(n) also gives the same thing but for avoiding any one of (-1,-2), (+2,+1) or (+1,+2) instead of (-2,-1) (see the Hardt and Troyka reference). - Justin M. Troyka, Aug 05 2011
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LINKS
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A. Hardt and J. M. Troyka, Slides (associated with the Hardt and Troyka reference above).
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FORMULA
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a(n) = Sum_{k=0..n-1} 2^{n-1-k}*[(n-1)! ]^2/[(k!)^2*(n-1-k)! ]. - Huajun Huang (huanghu(AT)auburn.edu), Oct 10 2005
a(0) = 1; a(1) = 3; a(n) = (4n-1) * a(n-1) - 4 (n-1)^2 * a(n-2) for n >= 2. - Justin M. Troyka, Aug 05 2011
E.g.f.: exp(2*x) * Sum_{n>=0} x^n/n!^2 = Sum_{n>=0} a(n)*x^n/n!^2. - Paul D. Hanna, Nov 18 2011
a(n) ~ n^(n+1/4)*2^(n-1/4)*exp(-n+sqrt(2*n)-1/4) * (1 + sqrt(2)/(3*sqrt(n))). - Vaclav Kotesovec, Jun 22 2013
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EXAMPLE
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Since a(2) = 17, there are 17 signed permutations of 4 that are equal to their reverse-complements and avoid (-2,-1). Some of these are: (+1,+3,+2,+4), (+2,-1,-4,+3), (+3,-1,-4,+2), (-1,-2,-3,-4). - Justin M. Troyka, Aug 05 2011
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MAPLE
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a := n -> (-2)^n*KummerU(-n, 1, -1/2):
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MATHEMATICA
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Table[ n! 2^n LaguerreL[ n, -1/2 ], {n, 0, 12} ]
f[n_] := Sum[k!*2^k*Binomial[n, k]^2, {k, 0, n}]; Table[ f[n], {n, 0, 17}] (* Robert G. Wilson v, Mar 16 2005 *)
a = {1, 3}; For[n = 2, n < 13, n++, a = Append[a, (4 n - 1) a[[n]] - 4 (n - 1)^2 a[[n - 1]]]]; a (* Justin M. Troyka, Aug 05 2011 *)
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PROG
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(PARI) {a(n)=n!^2*polcoeff(exp(2*x+x*O(x^n))*sum(m=0, n, x^m/m!^2), n)}
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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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