1,2

Number of Joyce trees with 2n-1 nodes. Number of tremolo permutations of {0,1,...,2n}. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Mar 28 2003

The Hankel transform of this sequence is A000178(n) for n odd = 1, 12, 34560, ...; example : det([1, 2, 16; 2, 16, 272, 16, 272, 7936]) = 34560 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 07 2004

a(n) = number of increasing labeled full binary trees with 2n-1 vertices. Full binary means every non-leaf vertex has two children, distinguished as left and right; labeled means the vertices are labeled 1,2,...,2n-1; increasing means every child has a label greater than its parent. - David Callan (callan(AT)stat.wisc.edu), Nov 29 2007

Contribution from Micha Hofri (hofri(AT)wpi.edu), May 27 2009: (Start)

a(n) was found to be the number of permutations of [2n] which when inserted in

order, to form a binary search tree, yield the maximally full possible tree (with only one single-child node).

The egf is sec^2(x)=1+tan^2(x), and the same coefficients can be manufactured from the tan(x) itself,

which is the egf for the number of trees as above for odd number of nodes. (End)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.

H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 69.

D. Dumont, Interpretations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.

Dominique Foata and Guo-Niu Han, Dimers and new q-tangent numbers, Preprint, 2008.

Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.

Knuth, D. E.; Buckholtz, Thomas J.; Computation of tangent, Euler and Bernoulli numbers. Math. Comp. 21 1967 663-688.

L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 148 (the numbers |C^{2n-1}| ).

J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, p. 282.

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

H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1, p. 20.

D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699.

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).

N. J. A. Sloane, The first 100 tangent numbers: Table of n, a(n) for n = 1..100

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

F. C. S. Brown, T. M. A. Fink and K. Willbrand, On arithmetic and asymptotic properties of up-down numbers

K.-W. Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 144

M.-P. Grosset and A. P. Veselov, Bernoulli numbers and solitons

A. R. Kr\"auter, Permanenten - Ein kurzer \"Uberblick

A. R. Kr\"auter, \"Uber die Permanente gewisser zirkul\"arer Matrizen...

N. E. Noerlund, Vorlesungen ueber Differenzenrechnung Springer 1924, p. 27.

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

R. Street, [math/0303267] Trees, permutations and the tangent function.

Ross Street, Trees, permutations and the tangent function gives definition of Joyce trees and tremolo permutations.

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

Eric Weisstein's World of Mathematics, Alternating Permutation

Index entries for "core" sequences

Index entries for sequences related to boustrophedon transform

Index entries for sequences related to Bernoulli numbers.

E.g.f.: log(sec x) = Sum_{n > 0} a(n)*x^(2*n)/(2*n)!.

E.g.f.: tan x = Sum_{n >= 0} a(n+1)*x^(2*n+1)/(2*n+1)!.

E.g.f.: (sec x)^2 = Sum_{n >= 0} a(n+1)*x^(2*n)/(2*n)!.

2/(exp(2x)+1) = 1 + Sum_{n>=1} (-1)^(n+1) a(n) x^(2n-1)/(2n-1)! = 1 - x + x^3/3 - 2*x^5/15 + 17*x^7/315 - 62*x^9/2835 + ...

a(n) = 2^(2*n) (2^(2*n) - 1) |B_(2*n)| / (2*n) where B_n are the Bernoulli numbers (A000367/A002445 or A027641/A027642).

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

Sum[2^(2*n + 1 - k)*(-1)^(n + k + 1)*k!*StirlingS2[2*n + 1, k], {k, 1, 2*n + 1}]. - Victor Adamchik, Oct 05 2005

a(n) = abs[c(2*n-1)] where c(n)= 2^(n+1) * (1-2^(n+1)) * Ber(n+1)/(n+1) = 2^(n+1) * (1-2^(n+1)) * (-1)^n * Zeta(-n) = [ -(1+EN(.))]^n = 2^n * GN(n+1)/(n+1) = 2^n * EP(n,0) = (-1)^n * E(n,-1) = (-2)^n * n! * Lag[n,-P(.,-1)/2] umbrally = (-2)^n * n! * C{T[.,P(.,-1)/2] + n, n} umbrally for the signed Euler numbers EN(n), the Bernoulli numbers Ber(n), the Genocchi numbers GN(n), the Euler polynomials EP(n,t), the Eulerian polynomials E(n,t), the Touchard / Bell polynomials T(n,t), the binomial function C(x,y) = x!/[(x-y)!*y! ] and the polynomials P(j,t) of A131758. - Tom Copeland (tcjpn(AT)msn.com), Oct 05 2007

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 27 2009: (Start)

a(1) = A094665(0,0)*A156919(0,0) and a(n) = sum(2^(n-k-1)*A094665(n-1, k)*A156919(k,0), k = 1..n-1) for n = 2, 3, .. , see A162005.

(End)

tan(x) = x + 2 x^3/3! + 16 x^5/5! + 272 x^7/7! + ... = x+1/3*x^3+2/15*x^5+17/315*x^7+62/2835*x^9+O(x^11).

tanh(x) = x-1/3*x^3+2/15*x^5-17/315*x^7+62/2835*x^9-1382/155925*x^11+...

(sec x)^2 = 1 + x^2 + 2/3*x^4 + 17/45*x^6 + ...

series(tan(x), x, 40);

with(numtheory): a := n-> abs(2^(2*n)*(2^(2*n)-1)*bernoulli(2*n)/(2*n));

Table[ Sum[2^(2*n + 1 - k)*(-1)^(n + k + 1)*k!*StirlingS2[2*n + 1, k], {k, 1, 2*n + 1}], {n, 0, 7}] - Victor Adamchik, Oct 05 2005

v[1] = 2; v[n_] /; n >= 2 := v[n] = Sum[ Binomial[2 n - 3, 2 k - 2] v[k] v[n - k], {k, n - 1}]; Table[ v[n]/2, {n, 15}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 08 2009]

(PARI) a(n)=if(n<1, 0, ((-4)^n-(-16)^n)*bernfrac(2*n)/2/n)

(PARI) a(n)=local(an); if(n<1, n>=0, an=vector(n+1, m, 1); for(m=1, n, an[m+1]=sum(k=0, m-1, binomial(2*m, 2*k+1)*an[k+1]*an[m-k])); an[n+1]) (from Michael Somos)

(PARI) a(n)=if(n<0, 0, (2*n+1)!*polcoeff(tan(x+O(x^(2*n+2))), 2*n+1)) (from Michael Somos)

a(n)=2^(n-1)*A002105(n). Apart from signs, 2^(2n-2)*A001469(n) = n*a(n).

Cf. A001469, A002430, A036279, A000364 (secant numbers), A000111 (secant-tangent numbers), A024283, A009764. First diagonal of A059419 and of A064190.

Cf. A009006, A009725, A029584, A012509, A009123, A009567.

Equals A002425(n) * 2^A101921(n).

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 27 2009: (Start)

Equals first left hand column of A162005.

(End)

Adjacent sequences: A000179 A000180 A000181 this_sequence A000183 A000184 A000185

Sequence in context: A050974 A012188 A009764 this_sequence A102599 A123744 A136796

nonn,core,easy,nice

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