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%I A000182 M2096 N0829
%S A000182 1,2,16,272,7936,353792,22368256,1903757312,209865342976,
%T A000182 29088885112832,4951498053124096,1015423886506852352,246921480190207983616,
%U A000182 70251601603943959887872,23119184187809597841473536
%N A000182 Tangent (or "Zag") numbers: expansion of tan x. Also expansion of tanh(x).
%C A000182 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
%C A000182 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
%C A000182 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
%C A000182 Contribution from Micha Hofri (hofri(AT)wpi.edu), May 27 2009: (Start)
%C A000182 a(n) was found to be the number of permutations of [2n] which when inserted 
               in
%C A000182 order, to form a binary search tree, yield the maximally full possible 
               tree (with only one single-child node).
%C A000182 The egf is sec^2(x)=1+tan^2(x), and the same coefficients can be manufactured 
               from the tan(x) itself,
%C A000182 which is the egf for the number of trees as above for odd number of nodes. 
               (End)
%D A000182 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
%D A000182 H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 
               1965, p. 69.
%D A000182 D. Dumont, Interpretations combinatoires des nombres de Genocchi, Duke 
               Math. J., 41 (1974), 305-318.
%D A000182 Dominique Foata and Guo-Niu Han, Dimers and new q-tangent numbers, Preprint, 
               2008.
%D A000182 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.
%D A000182 Knuth, D. E.; Buckholtz, Thomas J.; Computation of tangent, Euler and 
               Bernoulli numbers. Math. Comp. 21 1967 663-688.
%D A000182 L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 148 (the 
               numbers |C^{2n-1}| ).
%D A000182 J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, 
               p. 282.
%D A000182 S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; 
               see p. 444.
%D A000182 H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 
               1, p. 20.
%D A000182 D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 
               689-694; 22 (1968), 699.
%D A000182 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000182 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000182 N. J. A. Sloane, <a href="b000182.txt">The first 100 tangent numbers: 
               Table of n, a(n) for n = 1..100</a>
%H A000182 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 A000182 F. C. S. Brown, T. M. A. Fink and K. Willbrand, <a href="http://arXiv.org/
               abs/math.CO/0607763">On arithmetic and asymptotic properties of up-down 
               numbers</a>
%H A000182 K.-W. Chen, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Algorithms for Bernoulli numbers and Euler numbers</a>, J. Integer 
               Sequences, 4 (2001), #01.1.6.
%H A000182 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
               Publications/books.html">Analytic Combinatorics</a>, 2009; see page 
               144
%H A000182 M.-P. Grosset and A. P. Veselov, <a href="http://arXiv.org/abs/math.GM/
               0503175">Bernoulli numbers and solitons</a>
%H A000182 A. R. Kr\"auter, <a href="http://www.mat.univie.ac.at/~slc/opapers/s09kraeu.html">
               Permanenten - Ein kurzer \"Uberblick</a>
%H A000182 A. R. Kr\"auter, <a href="http://www.mat.univie.ac.at/~slc/opapers/s11kraeu.html">
               \"Uber die Permanente gewisser zirkul\"arer Matrizen...</a>
%H A000182 N. E. Noerlund, <a href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/
               digbib.cgi?PPN373206070">Vorlesungen ueber Differenzenrechnung</a> 
               Springer 1924, p. 27.
%H A000182 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 A000182 R. Street, <a href="http://arXiv.org/abs/math.HO/0303267">[math/0303267] 
               Trees, permutations and the tangent function</a>.
%H A000182 Ross Street, <a href="http://front.math.ucdavis.edu/math.HO/0303267">
               Trees, permutations and the tangent function</a> gives definition 
               of Joyce trees and tremolo permutations.
%H A000182 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               TangentNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A000182 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               AlternatingPermutation.html">Alternating Permutation</a>
%H A000182 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000182 <a href="Sindx_Bo.html#boustrophedon">Index entries for sequences related 
               to boustrophedon transform</a>
%H A000182 <a href="Sindx_Be.html#Bernoulli">Index entries for sequences related 
               to Bernoulli numbers.</a>
%F A000182 E.g.f.: log(sec x) = Sum_{n > 0} a(n)*x^(2*n)/(2*n)!.
%F A000182 E.g.f.: tan x = Sum_{n >= 0} a(n+1)*x^(2*n+1)/(2*n+1)!.
%F A000182 E.g.f.: (sec x)^2 = Sum_{n >= 0} a(n+1)*x^(2*n)/(2*n)!.
%F A000182 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 + ...
%F A000182 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).
%F A000182 Asymptotics: a(n) ~ 2^(2*n+1)*(2*n-1)!/Pi^(2*n).
%F A000182 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
%F A000182 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
%F A000182 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 27 
               2009: (Start)
%F A000182 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.
%F A000182 (End)
%e A000182 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).
%e A000182 tanh(x) = x-1/3*x^3+2/15*x^5-17/315*x^7+62/2835*x^9-1382/155925*x^11+...
%e A000182 (sec x)^2 = 1 + x^2 + 2/3*x^4 + 17/45*x^6 + ...
%p A000182 series(tan(x),x,40);
%p A000182 with(numtheory): a := n-> abs(2^(2*n)*(2^(2*n)-1)*bernoulli(2*n)/(2*n));
%t A000182 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
%t A000182 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]
%o A000182 (PARI) a(n)=if(n<1,0,((-4)^n-(-16)^n)*bernfrac(2*n)/2/n)
%o A000182 (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)
%o A000182 (PARI) a(n)=if(n<0,0,(2*n+1)!*polcoeff(tan(x+O(x^(2*n+2))),2*n+1)) (from 
               Michael Somos)
%Y A000182 a(n)=2^(n-1)*A002105(n). Apart from signs, 2^(2n-2)*A001469(n) = n*a(n).
%Y A000182 Cf. A001469, A002430, A036279, A000364 (secant numbers), A000111 (secant-tangent 
               numbers), A024283, A009764. First diagonal of A059419 and of A064190.
%Y A000182 Cf. A009006, A009725, A029584, A012509, A009123, A009567.
%Y A000182 Equals A002425(n) * 2^A101921(n).
%Y A000182 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 27 
               2009: (Start)
%Y A000182 Equals first left hand column of A162005.
%Y A000182 (End)
%Y A000182 Sequence in context: A050974 A012188 A009764 this_sequence A102599 A123744 
               A136796
%Y A000182 Adjacent sequences: A000179 A000180 A000181 this_sequence A000183 A000184 
               A000185
%K A000182 nonn,core,easy,nice
%O A000182 1,2
%A A000182 N. J. A. Sloane (njas(AT)research.att.com).

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