%I #16 Apr 19 2015 15:42:06

%S 4,18,88,460,2544,14776,89632,565392,3695680,24959776,173752704,

%T 1244125888,9146568448,68933546880,531838104064,4195358822656,

%U 33800254620672,277843218452992,2328182040156160,19870770461838336,172610363453599744,1525013813211609088

%N Number of irreducible representations of Sp(2n,R) with same infinitesimal character as the trivial representation.

%H Vincenzo Librandi, <a href="/A127394/b127394.txt">Table of n, a(n) for n = 1..200</a>

%H J. Adams and F. du Cloux, <a href="http://arxiv.org/abs/math/0701166">Algorithms for representation theory of real reductive groups</a>, arXiv:math/0701166.

%F E.g.f.: exp(4*x+x^2). - corrected by _Vaclav Kotesovec_, Oct 19 2012

%F E.g.f. 2*(x+2)*(1 + (x+4)*x/(G(0)-x^2-4*x)) where G(k)= x^2 + 4*x + k + 1 - (x+4)*x*(k+1)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Jul 12 2012

%F Recurrence: a(n) = 4*a(n-1) + 2*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 19 2012

%F a(n) ~ 2^(n/2-1/2)*exp(2*sqrt(2*n)-n/2-2)*n^(n/2)*(1+7/6*sqrt(2)/sqrt(n)). - _Vaclav Kotesovec_, Oct 19 2012

%t Rest[CoefficientList[Series[E^(4*x+x^2), {x, 0, 20}], x]* Range[0, 20]!] (* _Vaclav Kotesovec_, Oct 19 2012 *)

%o (PARI) x='x+O('x^66); Vec(serlaplace(2*(x+2)*exp(x*(x+4)))) /* _Joerg Arndt_, Jul 12 2012 */

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Apr 01 2007