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A117158
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Number of permutations avoiding the consecutive pattern 1234.
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13
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1, 1, 2, 6, 23, 111, 642, 4326, 33333, 288901, 2782082, 29471046, 340568843, 4263603891, 57482264322, 830335952166, 12793889924553, 209449977967081, 3630626729775362, 66429958806679686, 1279448352687538463
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OFFSET
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0,3
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COMMENTS
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a(n) is the number of permutations on [n] that avoid the consecutive pattern 1234. It is the same as the number of permutations which avoid 4321.
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LINKS
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Ray Chandler, Table of n, a(n) for n = 0..60
A. Baxter, B. Nakamura, and D. Zeilberger. Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes
Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, Adv. Appl. Math. 36 (2006) 138-155.
Sergi Elizalde and Marc Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003) 110-125.
I. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory A 53 (1990), 257-285.
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FORMULA
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E.g.f.: 2/(exp(-x)+cos(x)-sin(x)) = 1/W(0) with continued fraction
W(k) = 1+(x^(2*k))/(f+f*x/(4*k+1-x-(4*k+1)*b/R)) where R := x^(2*k)+b-(x^(4*k+1))/(c+(x^(2*k+1))+x*c/T); T := 4*k+3-x-(4*k+3)*d/(d+(x^(2*k+1))/W(k+1));
f:=(4*k)!/(2*k)!; b:=(4*k+1)!/(2*k+1)!; c:=(4*k+2)!/(2*k+1)!; d:=(4*k+3)!/(2*k+2)!. - Sergei N. Gladkovskii, Nov 30 2011
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MATHEMATICA
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a[n_]:=Coefficient[Series[2/(Cos[x]-Sin[x]+Exp[ -x]), {x, 0, 30}], x^n]*n!
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CROSSREFS
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Cf. A022558, A049774, A111004, A113228, A113229, A117156, A117226, A177523, A177533, A201692, A201693.
Sequence in context: A113228 A201693 A063255 * A185334 A059513 A132647
Adjacent sequences: A117155 A117156 A117157 * A117159 A117160 A117161
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KEYWORD
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nonn
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AUTHOR
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Steven Finch, Apr 26 2006
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STATUS
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approved
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