|
%I #30 Jul 20 2019 07:15:09
%S 1,2,5,14,44,152,569,2270,9524,41576,187348,866296,4092400,19684576,
%T 96156649,476038222,2384463044,12067926920,61641751124,317469893176,
%U 1647261806128,8605033903456,45228349510660,239061269168056,1270130468349904,6780349241182112,36355025167014224,195725149445320160,1057729059593103808
%N Moments of the quadratic coefficient of the characteristic polynomial of a random matrix in SU(2) X SU(2) (inside USp(4)).
%H Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, <a href="http://arxiv.org/abs/1110.6638">Sato-Tate distributions and Galois endomorphism modules in genus 2</a>, arXiv preprint arXiv:1110.6638 [math.NT], 2011-2012 (the sequence c-hat in Section 5.1.1).
%F a(n) = Sum_{k=0..n} binomial(n, k)*2^(n-k)*c(k)^2, where c() = A126120().
%F Conjecture: (n+2)^2*a(n) +2*(-3*n^2-5*n-1)*a(n-1) -4*(n-1)*(n-5)*a(n-2) +24*(n-1)*(n-2)*a(n-3)=0. - _R. J. Mathar_, Dec 04 2013 [ Maple's sumrecursion command applied to the above formula for a(n) produces this recurrence. - _Peter Bala_, Jul 06 2015 ]
%F a(n) ~ 2^(n-1) * 3^(n+3) / (Pi * n^3). - _Vaclav Kotesovec_, Jul 20 2019
%p b:=n->coeff((x^2+1)^n, x, n); # A126869
%p c:=n->b(n)/((n/2)+1); # A126120
%p ch:=n->add(binomial(n, k)*2^(n-k)*c(k)^2, k=0..n); # A202856
%p [seq(ch(n), n=0..30)];
%t b[n_] := Coefficient[(x^2+1)^n, x, n]; (* A126869 *)
%t c[n_] := b[n]/(n/2+1); (* A126120 *)
%t ch[n_] := Sum[Binomial[n, k] 2^(n-k) c[k]^2, {k, 0, n}]; (* A202856 *)
%t Table[ch[n], {n, 0, 30}] (* _Jean-François Alcover_, Aug 10 2018, translated from Maple *)
%Y Cf. A126869, A126120, A202814.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Dec 25 2011
|
