login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A013989 a(n) = (n+1)*(a(n-1)/n + a(n-2)), with a(0)=1, a(1)=2. 7

%I

%S 1,2,6,16,50,156,532,1856,6876,26200,104456,428352,1821976,7959056,

%T 35857200,165592576,785514512,3812387616,18948962656,96194028800,

%U 498931946016,2638959243712,14234346694976

%N a(n) = (n+1)*(a(n-1)/n + a(n-2)), with a(0)=1, a(1)=2.

%C 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]

%C 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]

%C 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

%D rec.puzzles, Dec 10 1995

%H Vincenzo Librandi, <a href="/A013989/b013989.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f: (1+x+x^2)*exp((1+x/2)*x). - _Benoit Cloitre_, Apr 28 2005, corrected by _Vaclav Kotesovec_, Oct 07 2012

%F a(n) = A000085(n) * (n+1).

%F a(n) = A000085(n+2) - A000085(n+1). - _M. F. Hasler_, Dec 26 2010

%F a(n) ~ n*exp(sqrt(n)-n/2-1/4)*n^(n/2)/sqrt(2). - _Vaclav Kotesovec_, Oct 07 2012

%F E.g.f. simplifies to x*exp(x + x^2/2) if offset is 1. [_David Callan_, Nov 11 2012]

%F G.f.: T(0)/x^2 - 1/x^2, where T(k) = 1 - (k+1)*x^2/( (k+1)*x^2 - (1-x)^2/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 03 2013

%F a(n) = (n+1) * (-i/sqrt(2))^n * H_n(i/sqrt(2)), where H_n(x) is the Hermite polynomial. - _Vladimir Reshetnikov_, Nov 12 2016

%F 0 = a(n)*(+a(n+1) - 2*a(n+2) - 2*a(n+3) + a(n+4)) + a(n+1)*(+3*a(n+2) + a(n+3) - a(n+4)) + a(n+2)*(-2*a(n+2) + a(n+3)) for all n in Z. - _Michael Somos_, Nov 12 2016

%e G.f. = 1 + 2*x + 6*x^2 + 16*x^3 + 50*x^4 + 156*x^5 + 532*x^6 + ...

%p A013989 := proc(n) option remember; if n <=1 then n+1; else (n+1)*(A013989(n-1)/n+A013989(n-2)); fi; end;

%t Table[n!*SeriesCoefficient[(1+x+x^2)*E^((1+x/2)*x),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 07 2012 *)

%t Table[(n + 1) (-I/Sqrt[2])^n HermiteH[n, I/Sqrt[2]], {n, 0, 30}] (* _Vladimir Reshetnikov_, Nov 12 2016 *)

%o (PARI) x='x+O('x^66); Vec(serlaplace((1+x+x^2)*exp((1+x/2)*x))) \\ _Joerg Arndt_, May 04 2013

%o (PARI) {a(n) = if( n<0, 0, n! * polcoeff( (1 + x + x^2) * exp( x * (1 + x/2 + O(x^n))), n))}; /* _Michael Somos_, Nov 12 2016 */

%Y First differences of A000085 (except for a missing leading zero).

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, _Dan Hoey_, 1996

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 12 09:36 EST 2019. Contains 329953 sequences. (Running on oeis4.)