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A025167
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E.g.f: exp(x/(1-2*x))/(1-2*x).
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4
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1, 3, 17, 139, 1473, 19091, 291793, 5129307, 101817089, 2250495523, 54780588561, 1455367098923, 41888448785857, 1298019439099059, 43074477771208913, 1523746948247663611, 57229027745514785793, 2274027983943883110467
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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 (troykaj(AT)carleton.edu), Aug 5 2011.
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REFERENCES
| A. Hardt and J. M. Troyka, Restricted symmetric signed permutations, pre-print. [For more information, email troykaj(AT)carleton.edu.]
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FORMULA
| Sum_{k=0..n} k!*3^k*C(n, k) (from Robert G. Wilson v Mar 16 2005)
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 (troykaj(AT)carleton.edu), Aug 5 2011.
E.g.f.: exp(2*x) * Sum_{n>=0} x^n/n!^2 = Sum_{n>=0} a(n)*x^n/n!^2. [From Paul D. Hanna, Nov 18 2011]
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EXAMPLE
| 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 (troykaj(AT)carleton.edu), Aug 5 2011.
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MATHEMATICA
| 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}] (from 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 (from Justin M. Troyka, Aug 5 2011.)
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PROG
| (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
| Cf. A025166, A025168, A002720.
Sequence in context: A199138 A006290 A060003 * A136727 A062873 A120022
Adjacent sequences: A025164 A025165 A025166 * A025168 A025169 A025170
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KEYWORD
| nonn
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AUTHOR
| w.meeussen (wouter.meeussen(AT)pandora.be)
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 29 2003
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