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

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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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

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

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)

(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.

Adjacent sequences: A000108 A000109 A000110 this_sequence A000112 A000113 A000114

Sequence in context: A009736 A104858 A138265 this_sequence A007976 A058259 A033543

nonn,core,eigen,nice,easy

njas