%I A000111 M1492 N0587
%S A000111 1,1,1,2,5,16,61,272,1385,7936,50521,353792,2702765,22368256,
%T A000111 199360981,1903757312,19391512145,209865342976,2404879675441,
%U A000111 29088885112832,370371188237525,4951498053124096,69348874393137901
%N A000111 Euler or up/down numbers: expansion of sec x + tan x . Also number of 
               alternating permutations on n letters.
%C A000111 Number of linear extensions of the "zig-zag" poset. See ch. 3, prob. 
               23 of Stanley. - Mitch Harris (Harris.Mitchell (AT) mgh.harvard.edu), 
               Dec 27, 2005
%C A000111 a(n) = number of increasing 0-1-2 trees on n vertices. - David Callan 
               (callan(AT)stat.wisc.edu), Dec 22 2006
%C A000111 Also the number of refinement of partitions. - Heinz-Richard Halder (halder.bichl(AT)t-online.de), 
               Mar 07 2008
%C A000111 Contribution from Pietro Majer (majer(AT)dm.unipi.it), Jul 13 2009: (Start)
%C A000111 The ratio a(n)/n! is also the probability that n numbers x1,x2,..,xn 
               randomly
%C A000111 choosen uniformly and independently in [0,1] satisfy x1>x2<x3>x4<...xn. 
               (End)
%D A000111 D. Andre', Sur les permutations alterne'es, Journal de Math\'{e}matiques 
               Pures et Appliqu\'{e}es, 7 (1881), 167-184.
%D A000111 Arnold, V. I., Bernoulli-Euler updown numbers associated with function 
               singularities, their combinatorics and arithmetics, Duke Math. J. 
               63 (1991), 537-555.
%D A000111 M. D. Atkinson: Zigzag permutations and comparisons of adjacent elements, 
               Information Processing Letters 21 (1985), 187-189.
%D A000111 M. D. Atkinson: Partial orders and comparison problems, Sixteenth Southeastern 
               Conference on Combinatorics, Graph Theory and Computing, (Boca Raton, 
               Feb 1985), Congressus Numerantium 47, 77-88.
%D A000111 L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 258-260, section 
               #11.
%D A000111 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied 
               Tables, Cambridge, 1966, p. 262.
%D A000111 H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 
               1965, p. 66.
%D A000111 N. D. Elkies, On the sums Sum_{k = -infinity .. infinity} (4k+1)^(-n), 
               Amer. Math. Monthly, 110 (No. 7, 2003), 561-573.
%D A000111 D. Foata and M.-P. Schutzenberger, Nombres d'Euler et permutations alternantes, 
               in J. N. Srivastava et al., eds., A Survey of Combinatorial Theory 
               (North Holland Publishing Company, Amsterdam, 1973), pp. 173-187.
%D A000111 Heinz-Richard Halder, Ueber Verfeinerungen von Partitionen, Periodica 
               Mathematica Hungarica Vol. 12 (3), (1981), pp. 217-220
%D A000111 O. Heimo and A. Karttunen, Series help-mates in 8, 9 and 10 moves (Problems 
               2901, 2974-2976), Suomen Tehtavaniekat (Proceedings of the Finnish 
               Chess Problem Society) vol. 60, no. 2/2006, p. 75, 77.
%D A000111 L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 238.
%D A000111 G. Kreweras, Les preordres totaux compatibles avec un ordre partiel. 
               Math. Sci. Humaines No. 53 (1976), 5-30.
%D A000111 A. Mendes, A note on alternating permutations, Amer. Math. Monthly, 114 
               (2007), 437-440.
%D A000111 S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; 
               see p. 444.
%D A000111 F. Murtagh, "Counting dendrograms: a survey", Discrete Applied Mathematics, 
               7 (1984), 191-199.
%D A000111 E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, 
               p. 110.
%D A000111 C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 184.
%D A000111 Y. Sano, The principal numbers of K. Saito for the types A_l, D_l and 
               E_l, Discr. Math., 307 (2007), 2636-2642.
%D A000111 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000111 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000111 J. Staib, Trigonometric power series, Math. Mag., 49 (1976), 147-148.
%D A000111 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see 
               Problem 5.7.
%D A000111 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1997.
%D A000111 Zhi-Hong Sun, Congruences involving Bernoulli polynomials, Discr. Math., 
               308 (2007), 71-112.
%H A000111 N. J. A. Sloane, <a href="b000111.txt">The first 200 Euler numbers: Table 
               of n, a(n) for n = 0..199</a>
%H A000111 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A000111 J. L. Arregui, <a href="http://arXiv.org/abs/math.NT/0109108">Tangent 
               and Bernoulli numbers</a> related to Motzkin and Catalan numbers 
               by means of numerical triangles.
%H A000111 B. Bauslaugh and F. Ruskey, <a href="http://www.cs.uvic.ca/~fruskey/Publications/
               ">Generating alternating permutations lexicographically</a>, Nordisk 
               Tidskr. Informationsbehandling (BIT) 30 16-26 1990.
%H A000111 F. Bergeron, M. Bousquet-M\'{e}lou and S. Dulucq, <a href="http://www.lacim.uqam.ca/
               ~annales/volumes/19-2/PDF/139-151.pdf">Standard paths in the composition 
               poset</a>
%H A000111 David Callan, <a href="http://www.stat.wisc.edu/~callan/papersother/">
               A note on downup permutations and increasing 0-1-2 trees</a>
%H A000111 N. D. Elkies, <a href="http://arXiv.org/abs/math.CA/0101168">On the sums 
               Sum((4k+1)^(-n),k,-inf,+inf)</a>
%H A000111 N. D. Elkies, <a href="http://www.combinatorics.org/Volume_11/Abstracts/
               v11i2a4.html">New Directions in Enumerative Chess Problems</a>, The 
               Electronic Journal of Combinatorics, vol. 11(2), 2004.
%H A000111 P. Flajolet, S. Gerhold and B. Salvy, <a href="http://arXiv.org/abs/math.CO/
               05011379">On the non-holonomic character of logarithms, powers and 
               the n-th prime function</a>
%H A000111 J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: 
               the Boustrophedon on transform, J. Combin. Theory, 17A 44-54 1996 
               (<a href="http://www.research.att.com/~njas/doc/bous.txt">Abstract</
               a>, <a href="http://www.research.att.com/~njas/doc/bous.pdf">pdf</
               a>, <a href="http://www.research.att.com/~njas/doc/bous.ps">ps</a>
               ).
%H A000111 A. Randrianarivony and J. Zeng, <a href="http://igd.univ-lyon1.fr/home/
               zeng/public_html/paper/publication.html">Sur une extension des nombres 
               d'Euler et les records des permutations alternantes</a>, J. Combin. 
               Theory Ser. A 68 (1994), 68-99.
%H A000111 A. Randrianarivony and J. Zeng, <a href="http://igd.univ-lyon1.fr/home/
               zeng/public_html/paper/publication.html">Une famille des polynomes 
               qui interpole plusieurs suites...</a>, Adv. Appl. Math. 17 (1996), 
               1-26.
%H A000111 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">
               My favorite integer sequences</a>, in Sequences and their Applications 
               (Proceedings of SETA '98).
%H A000111 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers.html#queue">
               Queue problems revisited</a>, Suomen Tehtavaniekat (Proceedings of 
               the Finnish Chess Problem Society), vol. 59, no. 4 (2005), 193-203.
%H A000111 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               EulerZigzagNumber.html">Link to a section of The World of Mathematics 
               (1).</a>
%H A000111 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               AlternatingPermutation.html">Link to a section of The World of Mathematics 
               (2).</a>
%H A000111 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               EntringerNumber.html">Link to a section of The World of Mathematics 
               (3).</a>
%H A000111 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000111 <a href="Sindx_Bo.html#boustrophedon">Index entries for sequences related 
               to boustrophedon transform</a>
%F A000111 a(n) = P_n(0) + Q_n(0) (see A155100 and A104035), defining Q_{-1} = 0. 
               Cf. A156142.
%F A000111 2*a(n+1) = Sum_{k=0..n} binomial(n, k)*a(k)*a(n-k). E.g.f.: tan x + sec 
               x.
%F A000111 Asymptotics: a(n) ~ 2^(n+2)*n!/Pi^(n+1).
%F A000111 a(n) = (n-1)*a(n-1) - sum{i=2, n-2, (i-1)*E(n-1, i)}, where E are the 
               Entringer numbers A008280. - Jon Perry (perry(AT)globalnet.co.uk), 
               Jun 09 2003
%F A000111 E.g.f. for a(n+1) = 1/(cos(x/2)-sin(x/2))^2 = 1/(1-sin(x)) = d/dx(sec(x)+tan(x)).
%F A000111 G.f. A(x)=y satisfies 2y'=1+y^2. - Michael Somos Feb 03 2004
%F A000111 a(2*k) = (-1)^k euler(2k) and a(2k-1) = (-1)^(k-1)2^(2k)(2^(2k)-1) bernoulli(2k)/
               (2k) - C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 17 2005
%F A000111 O.g.f.: A(x) = 1+x/(1-x-x^2/(1-2*x-3*x^2/(1-3*x-6*x^2/(1-4*x-10*x^2/(1-... 
               -n*x-(n*(n+1)/2)*x^2/(1- ...)))))) (continued fraction). - Paul D. 
               Hanna (pauldhanna(AT)juno.com), Jan 17 2006
%F A000111 |a(n+1)-2*a(n)|=A000708(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Jan 13 2007
%F A000111 a(n) = 2^n|E(n,1/2)+E(n,1)| where E(n,x) are the Euler polynomials. [From 
               Peter Luschny (peter(AT)luschny.de), Jan 25 2009]
%F A000111 a(n) = 2^{n+2}*n!*S(n+1)/(Pi)^{n+1}, where S(n)=Sum(1/(4k+1)^n, k=-inf..inf) 
               (see the Elkies reference). [From Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Aug 17 2009]
%e A000111 Sequence starts 1,1,2,5,16,... since possibilities are {}, {A}, {AB}, 
               {ACB, BCA}, {ACBD, ADBC, BCAD, BDAC, CDAB}, {ACBED, ADBEC, ADCEB, 
               AEBDC, AECDB, BCAED, BDAEC, BDCEA, BEADC, BECDA, CDAEB, CDBEA, CEADB, 
               CEBDA, DEACB, DEBCA}, etc. - Henry Bottomley (se16(AT)btinternet.com), 
               Jan 17 2001
%p A000111 A000111 := n-> n!*coeff(series(sec(x)+tan(x),x,n+1), x, n);
%p A000111 s := series(sec(x)+tan(x), x, 100): A000111 := n-> n!*coeff(s, x, n);
%p A000111 A000111:=n->piecewise(n mod 2=1,(-1)^((n-1)/2)*2^(n+1)*(2^(n+1)-1)*bernoulli(n+1)/
               (n+1),(-1)^(n/2)*euler(n)):seq(A000111(n),n=0..30); A000111:=proc(n) 
               local k: k:=floor((n+1)/2): if n mod 2=1 then RETURN((-1)^(k-1)*2^(2*k)*(2^(2*k)-1)*bernoulli(2*k)/
               (2*k)) else RETURN((-1)^k*euler(2*k)) fi: end:seq(A000111(n),n=0..30); 
               (C. Ronaldo)
%p A000111 T := n -> 2^n*abs(euler(n,1/2)+euler(n,1)): [From Peter Luschny (peter(AT)luschny.de), 
               Jan 25 2009]
%p A000111 Contribution from Peter Luschny (peter(AT)luschny.de), Jul 29 2009: (Start)
%p A000111 S := proc(n,k) option remember; if k=0 then RETURN(`if`(n=0,1,0)) fi; 
               S(n,k-1)+S(n-1,n-k) end:
%p A000111 A000364 := n -> S(2*n,2*n);
%p A000111 A000182 := n -> S(2*n+1,2*n+1);
%p A000111 A000111 := n -> S(n,n); (End)
%p A000111 a := proc (n) options operator, arrow: 2^(n+2)*factorial(n)*(sum(1/(4*k+1)^(n+1), 
               k = -infinity .. infinity))/Pi^(n+1) end proc: 1, seq(a(n), n = 1 
               .. 22); [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 17 2009]
%o A000111 (PARI) a(n)=if(n<1,n==0,n--;n!*polcoeff(1/(1-sin(x+x*O(x^n))),n)) (from 
               Michael Somos)
%o A000111 (PARI) a(n)=local(v=[1],t);if(n<0,0, for(k=2,n+2,t=0;v=vector(k,i,if(i>
               1,t+=v[k+1-i])));v[2]) (from Michael Somos)
%o A000111 (PARI) a(n)=local(an); if(n<1, n>=0, an=vector(n+1,m,1); for(m=2,n, an[m+1]=sum(k=0,
               m-1, binomial(m-1,k)*an[k+1]*an[m-k])/2); an[n+1]) (from Michael 
               Somos)
%Y A000111 Cf. A000364 (secant numbers), A000182 (tangent numbers). See also A008280, 
               A008281, A008282, A010094, A059720 for related triangles.
%Y A000111 A diagonal of A008970.
%Y A000111 Cf. A109449 for an extension to an exponential Riordan array. [From Peter 
               Luschny (peter(AT)luschny.de), Jan 25 2009]
%Y A000111 First column of A162170. [From Mats Granvik (mats.granvik(AT)abo.fi), 
               Jun 27 2009]
%Y A000111 Sequence in context: A009736 A104858 A138265 this_sequence A163747 A007976 
               A058259
%Y A000111 Adjacent sequences: A000108 A000109 A000110 this_sequence A000112 A000113 
               A000114
%K A000111 nonn,core,eigen,nice,easy
%O A000111 0,4
%A A000111 N. J. A. Sloane (njas(AT)research.att.com).