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A129535
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Number of permutations of 1,...,n with at least one pair of adjacent consecutive entries (i.e. of the form k(k+1) or (k+1)k; n>=2).
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1
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2, 6, 22, 106, 630, 4394, 35078, 315258, 3149494, 34620010, 415222566, 5395737242, 75516784982, 1132471183626, 18115911832390, 307919970965434, 5541804787940598, 105282261866132138, 2105441434230129254
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,1
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COMMENTS
| Column 1 of A129534. a(n)=n! - A002464(n).
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REFERENCES
| R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.40.
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FORMULA
| G.f.=E(x)-E(x(1-x)/(1+x)), where E(x)=Sum(n!x^n, n>=0).
a(n)=n!-sum((-1)^(n-k)*sum(binomial(i+k-1,k-1)*binomial(k,n-i-k),i,0,n-k)*k!,k,1,n), n>0. [From Kruchinin Vladimir (kru(AT)ie.tusur.ru), Sep 08 2010]
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EXAMPLE
| a(4)=22 because 3142 and 2413 are the only permutations of 1,2,3,4 with no adjacent consecutive entries.
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MAPLE
| E:=x->sum(n!*x^n, n=0..35): G:=E(x)-E(x*(1-x)/(1+x)): Gser:=series(G, x=0, 30): seq(coeff(Gser, x, n), n=2..23);
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CROSSREFS
| Cf. A129534, A002464.
Sequence in context: A129815 A103941 A064643 * A014371 A111280 A095817
Adjacent sequences: A129532 A129533 A129534 * A129536 A129537 A129538
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), May 05 2007
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