0,4

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

a(n) = number of increasing 0-1-2 trees on n vertices. - David Callan (callan(AT)stat.wisc.edu), Dec 22 2006

Also the number of refinement of partitions. - Heinz-Richard Halder (halder.bichl(AT)t-online.de), Mar 07 2008

Contribution from Pietro Majer (majer(AT)dm.unipi.it), Jul 13 2009: (Start)

The ratio a(n)/n! is also the probability that n numbers x1,x2,..,xn randomly

choosen uniformly and independently in [0,1] satisfy x1>x2<x3>x4<...xn. (End)

D. Andre', Sur les permutations alterne'es, Journal de Math\'{e}matiques Pures et Appliqu\'{e}es, 7 (1881), 167-184.

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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.

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 258-260, section #11.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 262.

H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 66.

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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.

Heinz-Richard Halder, Ueber Verfeinerungen von Partitionen, Periodica Mathematica Hungarica Vol. 12 (3), (1981), pp. 217-220

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S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 444.

F. Murtagh, "Counting dendrograms: a survey", Discrete Applied Mathematics, 7 (1984), 191-199.

E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 110.

C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 184.

Y. Sano, The principal numbers of K. Saito for the types A_l, D_l and E_l, Discr. Math., 307 (2007), 2636-2642.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. Staib, Trigonometric power series, Math. Mag., 49 (1976), 147-148.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.7.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1997.

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Joerg Arndt, Fxtbook

J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles.

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David Callan, A note on downup permutations and increasing 0-1-2 trees

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N. D. Elkies, New Directions in Enumerative Chess Problems, The Electronic Journal of Combinatorics, vol. 11(2), 2004.

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Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (2).

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (3).

Index entries for "core" sequences

Index entries for sequences related to boustrophedon transform

a(n) = P_n(0) + Q_n(0) (see A155100 and A104035), defining Q_{-1} = 0. Cf. A156142.

2*a(n+1) = Sum_{k=0..n} binomial(n, k)*a(k)*a(n-k). E.g.f.: tan x + sec x.

Asymptotics: a(n) ~ 2^(n+2)*n!/Pi^(n+1).

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

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)).

G.f. A(x)=y satisfies 2y'=1+y^2. - Michael Somos Feb 03 2004

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

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

|a(n+1)-2*a(n)|=A000708(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 13 2007

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]

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]

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

A000111 := n-> n!*coeff(series(sec(x)+tan(x), x, n+1), x, n);

s := series(sec(x)+tan(x), x, 100): A000111 := n-> n!*coeff(s, x, n);

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)

T := n -> 2^n*abs(euler(n, 1/2)+euler(n, 1)): [From Peter Luschny (peter(AT)luschny.de), Jan 25 2009]

Contribution from Peter Luschny (peter(AT)luschny.de), Jul 29 2009: (Start)

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:

A000364 := n -> S(2*n, 2*n);

A000182 := n -> S(2*n+1, 2*n+1);

A000111 := n -> S(n, n); (End)

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]

(PARI) a(n)=if(n<1, n==0, n--; n!*polcoeff(1/(1-sin(x+x*O(x^n))), n)) (from Michael Somos)

(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)

(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)

Cf. A000364 (secant numbers), A000182 (tangent numbers). See also A008280, A008281, A008282, A010094, A059720 for related triangles.

A diagonal of A008970.

Cf. A109449 for an extension to an exponential Riordan array. [From Peter Luschny (peter(AT)luschny.de), Jan 25 2009]

First column of A162170. [From Mats Granvik (mats.granvik(AT)abo.fi), Jun 27 2009]

Sequence in context: A009736 A104858 A138265 this_sequence A163747 A007976 A058259

Adjacent sequences: A000108 A000109 A000110 this_sequence A000112 A000113 A000114

nonn,core,eigen,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).