0,3

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

Number of permutations avoiding the patterns {2413, 3142, 2143}. - Vince Vatter (vince(AT)mcs.st-and.ac.uk), Aug 16 2006

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

Hankel transform of this sequence gives A000012 = [1,1,1,1,1,1,...] . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 24 2007

Starting with offset 1, Hankel transform = odd indexed Fibonacci numbers. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 27 2008]

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]

Equals INVERTi transform of A007317: (1, 2, 5, 15, 51, 188,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2009]

M. Albert, R. Aldred, M. Atkinson, C Handley, D. Holton, D. McCaughan and H. van Ditmarsch, Sorting Classes, Elec. J. of Comb. 12 (2005)

E. Deutsch, E. Munarini and S. Rinaldi, Skew Dyck paths (in preparation).

T. D. Noe, Table of n, a(n) for n = 0..200

J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

N. J. A. Sloane, Transforms

Index entries for reversions of series

R. Brignall, S. Huczynska and V. Vatter, Simple permutations and algebraic generating functions, arXiv:math.CO/0608391.

Also REVERT transform of x*(2*x-1)/(x^2+x-1) (Olivier Gerard).

G.f.: 2/(1 + x + sqrt(1 - 6x + 5x^2))

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

Binomial transform of Fine's sequence: a(n)=sum_{k=0..n} binomial (n, k)*A000957(n-k).

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]

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;

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 *)

Cf. A000957.

Cf. A128714.

Cf. A002212, A007317. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 19 2009]

Sequence in context: A111279 A150197 A150198 this_sequence A050203 A112806 A150199

Adjacent sequences: A033318 A033319 A033320 this_sequence A033322 A033323 A033324

nonn

Emeric Deutsch (deutsch(AT)duke.poly.edu)

More terms from Robert G. Wilson v (rgwv(at)rgwv.com), Nov 04 2005

Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Aug 07 2006