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A013989
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a(n) = (n+1)*( a(n-1)/n + a(n-2) ), with a(0)=1, a(1)=2.
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6
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1, 2, 6, 16, 50, 156, 532, 1856, 6876, 26200, 104456, 428352, 1821976, 7959056, 35857200, 165592576, 785514512, 3812387616, 18948962656, 96194028800, 498931946016, 2638959243712, 14234346694976
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OFFSET
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0,2
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COMMENTS
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a(n) is also the number of fixed points in all involutions (= self-inverse permutations) of {1,2,...,n+1}. Example: a(2)=6 because the involutions of {1,2,3} are 1'2'3', 1'32, 32'1, and 213', containing 6 fixed points (marked). [Emeric Deutsch, May 28 2009]
a(n) is also the number of adjacent transpositions in all involutions (= self inverse permutations) of {1,2,...,n+2}. Example: a(2)=6 because the involutions of {1,2,3,4} are 1234, 124*3, 13*24, 1432, 2*134, 2*14*3, 3214, 3412, 4231, and 43*21, containing 6 adjacent transpositions (marked with *). [Emeric Deutsch, Jun 08 2009]
It might be more natural to shift the index by 1 and prefix a(0)=0, then this would be exactly the first differences of A000085, and satisfy a(n)=n for n<3, a(n)/n = a(n-1)/(n-1)-a(n-2). - M. F. Hasler, Dec 25 2010
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REFERENCES
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rec.puzzles, Dec 10 1995
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
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FORMULA
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E.g.f: (1+x+x^2)*exp((1+x/2)*x). - Benoit Cloitre, Apr 28 2005, corrected by Vaclav Kotesovec, Oct 07 2012
a(n) = A000085(n) * (n+1).
a(n) = A000085(n+2) - A000085(n+1). - M. F. Hasler, Dec 26 2010
a(n) ~ n*exp(sqrt(n)-n/2-1/4)*n^(n/2)/sqrt(2). - Vaclav Kotesovec, Oct 07 2012
E.g.f. simplifies to x*exp(x + x^2/2) if offset is 1. [David Callan, Nov 11 2012]
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MAPLE
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A013989 := proc(n) option remember; if n <=1 then n+1; else (n+1)*(A013989(n-1)/n+A013989(n-2)); fi; end;
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MATHEMATICA
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Table[n!*SeriesCoefficient[(1+x+x^2)*E^((1+x/2)*x), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 07 2012 *)
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PROG
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(PARI) x='x+O('x^66); Vec(serlaplace((1+x+x^2)*exp((1+x/2)*x))) \\ Joerg Arndt, May 04 2013
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CROSSREFS
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First differences of A000085 (except for a missing leading zero).
Sequence in context: A213429 A195645 A000136 * A002841 A136509 A100664
Adjacent sequences: A013986 A013987 A013988 * A013990 A013991 A013992
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, Dan Hoey (Hoey(AT)AIC.NRL.Navy.Mil), 1996
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STATUS
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approved
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