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A082971
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Number of permutations of {1,2,...,n} containing exactly 3 occurrences of the 132 pattern.
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2
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1, 14, 82, 410, 1918, 8657, 38225, 166322, 716170, 3059864, 12994936, 54924212, 231235054, 970347575, 4060697955, 16952812170, 70629116910, 293720506860, 1219498444500, 5055891511980, 20933654593020, 86571545598642
(list; graph; refs; listen; history; internal format)
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OFFSET
| 4,2
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COMMENTS
| a(n)=A138160(n,3).
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REFERENCES
| M. Bona, Permutations with one or two 132-subsequences, Discrete Math., 181, 1998, 267-274.
M. Bona, The number of permutations with exactly r 132-subsequences is P-recursive in the size, Adv. Appl. Math., 18, 1997, 510-522.
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LINKS
| T. Mansour and A. Vainshtein, Counting occurrences of 132 in a permutation
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FORMULA
| a(n)=(2*n-9)!/n!/6/(n-5)!*(n^6+51*n^5-407*n^4-99*n^3+7750*n^2-22416*n+20160)
a(n)=(n^6 + 51n^5 - 407n^4 - 99n^3 + 7750n^2 - 22416n + 20160)(2n-9)!/[6 n!(n-5)! for n>=5; a(4)=1. G.f.=(1/2)(P(x) + Q(x)/(1-4x)^(5/2), where P(x)=2x^3 - 5x^2 + 7x - 2, Q(x)=-22x^6 - 106x^5 + 292x^4 - 302x^3 + 135x^2 - 27x + 2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 27 2008
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EXAMPLE
| a(4)=1 because we have 1432 (the 132 occurrences are 143, 142 and 132).
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MAPLE
| P:=2*x^3-5*x^2+7*x-2: Q:=-22*x^6-106*x^5+292*x^4-302*x^3+135*x^2-27*x+2: g:= (P+Q/(1-4*x)^(5/2))*1/2: gser:=series(g, x=0, 30): seq(coeff(gser, x, n), n=4..25); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 27 2008
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PROG
| (PARI) a(n)=(2*n-9)!/n!/6/(n-5)!*(n^6+51*n^5-407*n^4-99*n^3+7750*n^2-22416*n+20160)
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CROSSREFS
| Cf. A002054, A082970, A138162, A138163, A138160.
Sequence in context: A138401 A099360 A199912 * A176010 A166819 A108683
Adjacent sequences: A082968 A082969 A082970 * A082972 A082973 A082974
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KEYWORD
| nonn
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AUTHOR
| Benoit Cloitre (benoit7848c(AT)orange.fr), May 27 2003
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), May 21 2008 at the suggestion of R. J. Mathar
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