%I A033321
%S A033321 1,1,2,6,21,79,311,1265,5275,22431,96900,424068,1876143,8377299,37704042,
%T A033321 170870106,779058843,3571051579,16447100702,76073821946,353224531663,
%U A033321 1645807790529,7692793487307,36061795278341,169498231169821
%N A033321 Binomial transform of Fine's sequence A000957: 1,0,1,2,6,18,57,186,...
%C A033321 Number of permutations avoiding the patterns {2431,4231,4321}; number 
               of weak sorting class based on 2431. - Len Smiley ( smiley (at) math.uaa.alaska.edu 
               ), Nov 01 2005
%C A033321 Number of permutations avoiding the patterns {2413, 3142, 2143}. - Vince 
               Vatter (vince(AT)mcs.st-and.ac.uk), Aug 16 2006
%C A033321 Number of skew Dyck paths of semilength n ending with a down step (1,
               -1). A skew Dyck path is a path in the first quadrant which begins 
               at the origin, ends on the x-axis, consists of steps U=(1,1)(up), 
               D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not 
               overlap. The length of the path is defined to be the number of its 
               steps. Number of skew Dyck paths of semilength n and ending with 
               a left step is A128714(n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               May 11 2007
%C A033321 Hankel transform of this sequence gives A000012 = [1,1,1,1,1,1,...] . 
               - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 24 2007
%C A033321 Starting with offset 1, Hankel transform = odd indexed Fibonacci numbers. 
               [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 27 2008]
%C A033321 Starting with offset 1 = INVERT transform of A002212: (1, 1, 3, 10, 36, 
               137,...) [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 19 2009]
%C A033321 Equals INVERTi transform of A007317: (1, 2, 5, 15, 51, 188,...). [From 
               Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2009]
%D A033321 M. Albert, R. Aldred, M. Atkinson, C Handley, D. Holton, D. McCaughan 
               and H. van Ditmarsch, Sorting Classes, Elec. J. of Comb. 12 (2005)
%D A033321 E. Deutsch, E. Munarini and S. Rinaldi, Skew Dyck paths (in preparation).
%H A033321 T. D. Noe, <a href="b033321.txt">Table of n, a(n) for n = 0..200</a>
%H A033321 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 
               4 (2001), #01.1.5.
%H A033321 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%H A033321 <a href="Sindx_Res.html#revert">Index entries for reversions of series</
               a>
%H A033321 R. Brignall, S. Huczynska and V. Vatter, <a href="http://arXiv.org/abs/
               math.CO/0608391">Simple permutations and algebraic generating functions</
               a>, arXiv:math.CO/0608391.
%F A033321 Also REVERT transform of x*(2*x-1)/(x^2+x-1) (Olivier Gerard).
%F A033321 G.f.: 2/(1 + x + sqrt(1 - 6x + 5x^2))
%F A033321 a(n)=[(13n-5)a(n-1)-(16n-23)a(n-2)+5(n-2)a(n-3)]/[2(n+1)] (n>=3); a[0]=a[1]=1, 
               a[2]=2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 21 2004
%F A033321 Binomial transform of Fine's sequence: a(n)=sum_{k=0..n} binomial (n, 
               k)*A000957(n-k).
%F A033321 G.f.: 1/(1-x-x^2/(1-3x-x^2/(1-3x-x^2/(1-3x-x^2/(1-... (continued fraction). 
               [From Paul Barry (pbarry(AT)wit.ie), Jun 15 2009]
%p A033321 a[0] := 1: a[1] := 1: a[2] := 2: for n from 3 to 23 do a[n] := ((13*n-5)*a[n-1]-(16*n-23)*a[n-2]+5*(n-2)*a[n-\
               3])/2/(n+1) od;
%t A033321 f[n_] := Sum[Binomial[n, k]*g[n - k], {k, 0, n}]; g[n_] := Sum[(-1)^(m 
               + n)(n + m)!/n!/m!(n - m + 1)/(n + 1), {m, 0, n}]; Table[ f[n], {n, 
               24}] (* Robert G. Wilson v *)
%Y A033321 Cf. A000957.
%Y A033321 Cf. A128714.
%Y A033321 Cf. A002212, A007317. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 
               19 2009]
%Y A033321 Sequence in context: A111279 A150197 A150198 this_sequence A050203 A112806 
               A150199
%Y A033321 Adjacent sequences: A033318 A033319 A033320 this_sequence A033322 A033323 
               A033324
%K A033321 nonn
%O A033321 0,3
%A A033321 Emeric Deutsch (deutsch(AT)duke.poly.edu)
%E A033321 More terms from Robert G. Wilson v (rgwv(at)rgwv.com), Nov 04 2005
%E A033321 Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Aug 07 2006