cp's OEIS Frontend

A065619 Expansion of e.g.f. x * (tan(x) + sec(x)).

%I A065619 #81 Aug 06 2024 04:36:41
%S A065619 1,2,3,8,25,96,427,2176,12465,79360,555731,4245504,35135945,313155584,
%T A065619 2990414715,30460116992,329655706465,3777576173568,45692713833379,
%U A065619 581777702256640,7777794952988025,108932957168730112,1595024111042171723,24370173276164456448
%N A065619 Expansion of e.g.f. x * (tan(x) + sec(x)).
%C A065619 a(n) is the number of down-up permutations w on [n+1] such that w_2 = 1. For example, a(3)=3 counts 2143, 3142, 4132. - _David Callan_, Oct 25 2004
%C A065619 A signed variant of this sequence, prefaced with an 0, is column 1 of the inverse of triangle A178616. - _Gary W. Adamson_, May 30 2010
%C A065619 a(n) is the number of ranked unlabeled binary tree shapes compatible with the binary perfect phylogeny (n,2). - _Noah A Rosenberg_, Jun 03 2022
%H A065619 Bishal Deb and Alan D. Sokal, <a href="https://arxiv.org/abs/2212.07232">Classical continued fractions for some multivariate polynomials generalizing the Genocchi and median Genocchi numbers</a>, arXiv:2212.07232 [math.CO], 2022. See p. 11.
%H A065619 S. Kitaev and T. Mansour, <a href="https://arxiv.org/abs/math/0205182">Simultaneous avoidance of generalized patterns</a>, arXiv:math/0205182 [math.CO], 2002.
%H A065619 H. D. Nguyen and D. Taggart, <a href="https://citeseerx.ist.psu.edu/pdf/8f2f36f22878c984775ed04368b8893879b99458">Mining the OEIS: Ten Experimental Conjectures</a>, 2013. Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014
%H A065619 J. A. Palacios, A. Bhaskar, F. Disanto and N. A. Rosenberg, <a href="https://doi.org/10.1007/s00285-022-01748-w">Enumeration of binary trees compatible with a perfect phylogeny</a>, J. Math. Biol. 84 (2022), 54.
%F A065619 E.g.f.: x*(tan(x)+sec(x)).
%F A065619 a(n) = n*A000111(n-1). - _R. J. Mathar_, Nov 27 2006
%F A065619 a(n) = n*2^(n-1)*|E_{n-1}(1/2)+E_{n-1}(1)| for n > 1; E_{n}(x) Euler polynomial. - _Peter Luschny_, Nov 25 2010
%F A065619 E.g.f.: x*(tan(x)+sec(x))=x+x^2/U(0);U(k)=4k+1-x/(2-x/(4k+3+x/(2+x/U(k+1))));(continued fraction). - _Sergei N. Gladkovskii_, Nov 14 2011
%F A065619 E.g.f.: x*( tan(x) + sec(x) )= x + 2*x^2/(U(0)-x) where U(k)= 4*k+2 - x^2/U(k+1);(continued fraction, 1-step). - _Sergei N. Gladkovskii_, Nov 07 2012
%F A065619 a(n) = (n + 1)!*Re([x^n](1 + i^((n - 1)*n)*(2 - 2*i)/(exp(x) + i))), assuming offset = 0. - _Peter Luschny_, Aug 09 2021
%F A065619 a(n) = 2*n*i^n*PolyLog(1 - n, -i) for n >= 2. - _Peter Luschny_, Aug 17 2021
%p A065619 A065619 := n -> `if`(n=1,1,2^(n-1)*abs(euler(n-1,1/2)+euler(n-1,1))*n): # _Peter Luschny_, Nov 25 2010
%p A065619 # Alternatively (after _Alois P. Heinz_):
%p A065619 b := proc(u, o) option remember;
%p A065619 `if`(u + o = 0, 1, add(b(o - 1 + j, u - j), j = 1..u)) end:
%p A065619 a := n -> n*b(n-1, 0): seq(a(n), n = 1..24); # _Peter Luschny_, Oct 27 2017
%t A065619 a[1] = 1; a[n_] := 2^(n-1)*Abs[EulerE[n - 1, 1/2] + EulerE[n - 1, 1]]*n; Array[a, 24] (* _Jean-François Alcover_, Nov 05 2017, after _Peter Luschny_ *)
%t A065619 Table[Re[2 n I^n PolyLog[1 - n, -I]], {n, 1, 19}] (* _Peter Luschny_, Aug 17 2021 *)
%o A065619 (PARI) {a(n) = if( n<0, 0 ,n! * polcoeff( x * (tan(x + x * O(x^n)) + 1 / cos(x + x * O(x^n))), n))}
%o A065619 (PARI)
%o A065619 x='x+O('x^66);
%o A065619 egf=x*(tan(x)+1/cos(x));
%o A065619 Vec(serlaplace(egf))
%o A065619 /* _Joerg Arndt_, May 28 2012 */
%o A065619 (Sage) # Algorithm of L. Seidel (1877)
%o A065619 def A065619_list(n) : # starts with a(0) = 0.
%o A065619     R = []; A = {-1:1, 0:0}; k = 0; e = 1
%o A065619     for i in (0..n) :
%o A065619         Am = 0; A[k + e] = 0; e = -e
%o A065619         for j in (0..i) : Am += A[k]; A[k] = Am; k += e
%o A065619         R.append(A[-i//2] if i%2 == 0 else A[i//2])
%o A065619     return R
%o A065619 A065619_list(22) # _Peter Luschny_, May 27 2012
%o A065619 (Python)
%o A065619 from itertools import accumulate
%o A065619 def A065619(n):
%o A065619     if n <= 2: return n
%o A065619     blist = (0,1)
%o A065619     for _ in range(n-2):
%o A065619         blist = tuple(accumulate(reversed(blist),initial=0))
%o A065619     return blist[-1]*n # _Chai Wah Wu_, Apr 25 2023
%Y A065619 Cf. A000111, A099028, A178616.
%K A065619 nonn
%O A065619 1,2
%A A065619 _Michael Somos_, Dec 03 2001