A007836 Springer numbers associated with symplectic group.

1, 1, 1, 5, 23, 151, 1141, 10205, 103823, 1190191, 15151981, 212222405, 3242472023, 53670028231, 956685677221, 18271360434605, 372221031054623, 8056751598834271, 184647141575344861, 4466900836910758805

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

Comments from F. Chapoton, Oct 30 2009: To compute this sequence, I used something similar to the Boustrophedon definition of the Euler numbers, but with two triangles instead of one. This is described (page 94) in Arnold's article in "Leçons de mathématiques d'aujourd'hui, volume 1" Editions Cassini. This is very similar to A001586, except that the initial conditions ( (0,1) at top of the two triangles ) are exchanged.

References

• V. I. Arnold, The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups, Uspekhi Mat. Nauk., 47 (#1, 1992), 3-45 = Russian Math. Surveys, Vol. 47 (1992), 1-51.
• V. I. Arnold, Nombres d'Euler, de Bernoulli et de Springer pour les groupes de Coxeter et les espaces de morsification : le calcul des serpents, in "Leçons de mathématiques d'aujourd'hui, volume 1", Editions Cassini.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Michael E. Hoffman, Derivative Polynomials, Euler Polynomials, and Associated Integer Sequences, The Electronic Journal of Combinatorics, Volume 6.1 (1999): Research paper R21, 13 p.
• M. Josuat-Verges, J.-C. Novelli and J.-Y. Thibon, The algebraic combinatorics of snakes, arXiv preprint arXiv:1110.5272 [math.CO], 2011.
• A. Vieru, Agoh's conjecture: its proof, its generalizations, its analogues, arXiv preprint arXiv:1107.2938 [math.NT], 2011-2012.

Crossrefs

Cf. A001586, A155100, A104035, A156142.

Programs

Mathematica

p[n_, u_] := D[Tan[x], {x, n}] /. Tan[x] -> u /. Sec[x] -> Sqrt[1+u^2] // Expand; p[-1, u_] = 1; t[n_, k_] := t[n, k] = k*t[n-1, k-1]+(k+1)*t[n-1, k+1]; t[0, 0] = 1; t[0, ] = 0; t[-1, ] = 0; q[n_, u_] := Sum[t[n, k]*u^k, {k, 0, n}]; a[n_] := p[n, 1]-q[n, 1]; a[0]=1; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Feb 05 2014 *)
nmax = 20; CoefficientList[Series[1 + (Sin[x] + Cos[x] - 1) / (Cos[x] - Sin[x]), {x, 0, nmax}], x] * Range[0,nmax]! (* Vaclav Kotesovec, Dec 08 2020 *)

Formula

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

From Vaclav Kotesovec, Dec 08 2020: (Start)

E.g.f.: (2*cos(x) - 1) / (cos(x) - sin(x)).

a(n) ~ (2 - sqrt(2)) * 2^(2*n + 3/2) * n^(n + 1/2) / (Pi^(n + 1/2) * exp(n)). (End)

Extensions

More terms from F. Chapoton, Oct 30 2009