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A033321
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Binomial transform of Fine's sequence A000957: 1,0,1,2,6,18,57,186,...
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17
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1, 1, 2, 6, 21, 79, 311, 1265, 5275, 22431, 96900, 424068, 1876143, 8377299, 37704042, 170870106, 779058843, 3571051579, 16447100702, 76073821946, 353224531663, 1645807790529, 7692793487307, 36061795278341, 169498231169821
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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]
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REFERENCES
| M. Albert, R. Aldred, M. Atkinson, C Handley, D. Holton, D. McCaughan and H. van Ditmarsch, Sorting Classes, Elec. J. of Comb. 12 (2005)
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LINKS
| T. D. Noe, Table of n, a(n) for n = 0..200
R. Brignall, S. Huczynska and V. Vatter, Simple permutations and algebraic generating functions, arXiv:math.CO/0608391.
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
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
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FORMULA
| 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(n)= Sum_[k, 0<=k<=n} A091965(n,k)*(-2)^k. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 28 2009]
a(n)=sum(m=1..n-1, sum(k=1..n-m, binomial(n-m-1,k-1)*m/(k+m)*binomial(2*k+m-1,k+m-1)))+1. [Vladimir Kruchinin kru(AT)ie.tusur.ru, May 12 2011]
a(n) = upper left term in M^n, M = the production matrix:
1, 1, 0, 0, 0, 0, 0,...
1, 2, 1, 0, 0, 0, 0,...
1, 2, 1, 1, 0, 0, 0,...
1, 2, 1, 2, 1, 0, 0,...
1, 2, 1, 2, 1, 1, 0,...
1, 2, 1, 2, 1, 2, 1,...
...
- Gary W. Adamson, Jul 08 2011
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MAPLE
| 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;
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MATHEMATICA
| 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 *)
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PROG
| (Maxima)
a(n):=sum(sum(binomial(n-m-1, k-1)*m/(k+m)*binomial(2*k+m-1, k+m-1), k, 1, n-m), m, 1, n-1)+1. [Vladimir Kruchinin kru(AT)ie.tusur.ru, May 12 2011]
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CROSSREFS
| Cf. A000957.
Cf. A128714.
Cf. A002212, A007317. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 19 2009]
Sequence in context: A111279 A150197 A150198 * A050203 A112806 A150199
Adjacent sequences: A033318 A033319 A033320 * A033322 A033323 A033324
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu)
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EXTENSIONS
| 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
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