A005981
Number of 2 up, 2 down, 2 up, ... permutations of length 2n + 1.
Original entry on oeis.org
1, 1, 6, 71, 1456, 45541, 2020656, 120686411, 9336345856, 908138776681, 108480272749056, 15611712012050351, 2664103110372192256, 531909061958526321421, 122840808510269863827456, 32491881630252866646683891, 9758611490955498257378246656
Offset: 0
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- P. R. Stein, personal communication.
- Alois P. Heinz, Table of n, a(n) for n = 0..100
- Nicolas Basset, Counting and generating permutations using timed languages, 2013.
- Nicolas Basset, Counting and generating permutations in regular classes of permutations, HAL Id: hal-01093994, 2014.
- B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, arXiv:math/0209062 [math.AG], 2002; Moscow Math. J., 3 (2003), 647-659.
- P. R. Stein & N. J. A. Sloane, Correspondence, 1975
- Eric Weisstein's World of Mathematics, Generalized Hyperbolic Functions
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g:=((cosh(x)-1)*sin(x)+(cos(x)+1)*sinh(x))/(cos(x)*cosh(x)+1): series(%,x,35):
seq(n!*coeff(%,x,n),n=1..34,2); # Peter Luschny, Feb 07 2017
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egf = ((Cosh[x]-1)*Sin[x]+(Cos[x]+1)*Sinh[x])/(Cos[x]*Cosh[x]+1); a[n_] := SeriesCoefficient[egf, {x, 0, 2*n+1}]*(2*n+1)!; Array[a, 17, 0] (* Jean-François Alcover, Mar 13 2014 *)
A131453
2 up, 2 down, ..., 2 up, 2 down permutations of length 4n+1.
Original entry on oeis.org
1, 6, 1456, 2020656, 9336345856, 108480272749056, 2664103110372192256, 122840808510269863827456, 9758611490955498257378246656, 1251231616578606273788469919481856, 245996119743058288132230759497577005056, 71155698830255977656506481145458378597728256
Offset: 0
a(1) = 6. The six 2 up, 2 down permutations on 5 letters are (12543), (13542), (14532), (23541), (24532) and (34521).
- Alois P. Heinz, Table of n, a(n) for n = 0..50
- L. Olivier, Bemerkungen über eine Art von Functionen, welche ähnliche Eigenschaften haben, wie der Cosinus und Sinus, J. Reine Angew. Math. 2 (1827), 243-251.
- H. Prodinger and T. A. Tshifhumulo, On q-Olivier Functions, Annals of Combinatorics 6 (2002), 181-194.
- B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, arXiv:math/0209062 [math.AG], 2002.
- B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, Moscow Math. J. 3 (2003), 647-659.
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g:= (tan(x)+exp(2*x)*(tan(x)+1)-1)/(exp(2*x)+2*exp(x)*sec(x)+1): series(%,x,46):
seq(n!*coeff(%,x,n), n=1..45,4); # Peter Luschny, Feb 07 2017
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Table[(CoefficientList[Series[((-1 + E^(2*x))*Cos[x] + (1 + E^(2*x))*Sin[x]) / (2*E^x + (1 + E^(2*x))* Cos[x]), {x, 0, 80}], x] * Range[0, 77]!)[[n]], {n, 2, 78, 4}] (* Vaclav Kotesovec, Sep 09 2014 *)
A131454
2 up, 2 down, ..., 2 up, 2 down, 2 up permutations of length 4n+3.
Original entry on oeis.org
1, 71, 45541, 120686411, 908138776681, 15611712012050351, 531909061958526321421, 32491881630252866646683891, 3302814916156503291298772711761, 527393971346575736206847604137659031, 126355819963625435928020023737689391659701
Offset: 0
(1 4 5 3 2 6 7) is a 2 up, 2 down, 2 up permutation - one of the seventy-one permutations of this type in the symmetric group on 7 letters.
- Alois P. Heinz, Table of n, a(n) for n = 0..50
- Christopher R. H. Hanusa, Alejandro H. Morales, and Martha Yip, Column convex matrices, G-cyclic orders, and flow polytopes, arXiv:2107.07326 [math.CO], 2021.
- L. Olivier, Bemerkungen über eine Art von Functionen, welche ähnliche Eigenschaften haben, wie der Cosinus und Sinus, J. Reine Angew. Math. 2 (1827), 243-251.
- H. Prodinger and T. A. Tshifhumulo, On q-Olivier Functions, Annals of Combinatorics 6 (2002), 181-194.
- B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, arXiv:math/0209062 [math.AG], 2002.
- B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, Moscow Math. J. 3 (2003), 647-659.
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g:=(sinh(x)-sin(x))/(cos(x)*cosh(x)+1): series(%,x,44):
seq(n!*coeff(%,x,n),n=3..45,4); # Peter Luschny, Feb 07 2017
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Table[(CoefficientList[Series[(-Sin[x] + Sinh[x]) / (1 + Cos[x]*Cosh[x]), {x, 0, 60}], x] * Range[0, 59]!)[[n]], {n, 4, 60, 4}] (* Vaclav Kotesovec, Sep 09 2014 *)
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