A127394 Number of irreducible representations of Sp(2n,R) with same infinitesimal character as the trivial representation.

4, 18, 88, 460, 2544, 14776, 89632, 565392, 3695680, 24959776, 173752704, 1244125888, 9146568448, 68933546880, 531838104064, 4195358822656, 33800254620672, 277843218452992, 2328182040156160, 19870770461838336, 172610363453599744, 1525013813211609088

Offset

1,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

J. Adams and F. du Cloux, Algorithms for representation theory of real reductive groups, arXiv:math/0701166.

Formula

E.g.f.: exp(4*x+x^2). - corrected by Vaclav Kotesovec, Oct 19 2012

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

Recurrence: a(n) = 4*a(n-1) + 2*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 19 2012

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

Mathematica

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

Prog

(PARI) x='x+O('x^66);  Vec(serlaplace(2*(x+2)*exp(x*(x+4)))) /* Joerg Arndt, Jul 12 2012 */

Crossrefs

Sequence in context: A260650 A006629 A068764 * A046984 A129323 A000305

Adjacent sequences: A127391 A127392 A127393 * A127395 A127396 A127397

Keyword

nonn

Author

N. J. A. Sloane, Apr 01 2007

Status

approved