%I A095922
%S A095922 1,0,1,0,3,1,15,15,105,190,945,2410,10263,31890,127699,444458,1751685,
%T A095922 6518736,25807445,100152288,401449271,1602902055,6519160851,26580508625,
%U A095922 109656966853,454524861846,1899821492925,7982263725826,33757439931675
%N A095922 Dimension of invariants of n-th tensor power of 5-dimensional irreducible 
               representation of B_2.
%C A095922 The analogous sequence for G_2 is A059710.
%D A095922 Alec Mihailovs, A Combinatorial Approach to Representations of Lie Groups 
               and Algebras, Springer-Verlag New York (2004).
%F A095922 a(n) =sum(A000108(i)*A000108(i+1)*binomial(n, 2*i), i=0..floor(n/2)) 
               - sum(A000108(i)^2*binomial(n, 2*i-1), i=0..floor((n+1)/2)); exponential 
               generating function = exp(t)*b(t) where b(t) is the exponential generating 
               function of the sequence B(n) = (-1)^n*A000108(floor((n+1)/2))*A000108(floor(n/
               2+1)).
%F A095922 a(0)=1, a(1)=0, a(2)=1 and (n+3)(n+4)a(n)=3(n-1)(n+2)a(n-1)+(n-1)(13n+4)a(n-2)-15(n-1)(n-2)a(n-3) 
               for n>2.
%e A095922 a(2)=1 because SO(5) has unique (up to multiplication by a constant) 
               invariant in V\otimes V - the quadratic form x^2+y^2+z^2+u^2+v^2.
%p A095922 ca:=n->binomial(n+n,n)/(n+1); a:=n->add(ca(i)*ca(i+1)*binomial(n,2*i),
               i=0..floor(n/2))- add(ca(i)^2*binomial(n,2*i-1),i=0..floor((n+1)/
               2)); seq(a(n),n=0..40);
%p A095922 A095922:=rsolve({(n+3)*(n+4)*A(n)=3*(n-1)*(n+2)*A(n-1)+(n-1)*(13*n+4)*A(n-2)-15*(n-1)*(n-2)*A(n-3),
               A(0)=1,A(1)=0,A(2)=1},A(n),makeproc);
%Y A095922 Cf. A000108, A059710.
%Y A095922 Sequence in context: A144006 A113378 A156289 this_sequence A089278 A087071 
               A053485
%Y A095922 Adjacent sequences: A095919 A095920 A095921 this_sequence A095923 A095924 
               A095925
%K A095922 easy,nice,nonn
%O A095922 0,5
%A A095922 Alec Mihailovs (alec(AT)mihailovs.com), Jul 11 2004