A202856 Moments of the quadratic coefficient of the characteristic polynomial of a random matrix in SU(2) X SU(2) (inside USp(4)).

1, 2, 5, 14, 44, 152, 569, 2270, 9524, 41576, 187348, 866296, 4092400, 19684576, 96156649, 476038222, 2384463044, 12067926920, 61641751124, 317469893176, 1647261806128, 8605033903456, 45228349510660, 239061269168056, 1270130468349904, 6780349241182112, 36355025167014224, 195725149445320160, 1057729059593103808

Offset: 0

Author

N. J. A. Sloane, Dec 25 2011

Keywords

• nonn

Links

• Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv preprint arXiv:1110.6638 [math.NT], 2011-2012 (the sequence c-hat in Section 5.1.1).

Crossrefs

Cf. A126869, A126120, A202814.

Programs

Maple

b:=n->coeff((x^2+1)^n, x, n); # A126869
c:=n->b(n)/((n/2)+1); # A126120
ch:=n->add(binomial(n, k)*2^(n-k)*c(k)^2, k=0..n); # A202856
[seq(ch(n), n=0..30)];

Mathematica

b[n_] := Coefficient[(x^2+1)^n, x, n]; (* A126869 *)
c[n_] := b[n]/(n/2+1); (* A126120 *)
ch[n_] := Sum[Binomial[n, k] 2^(n-k) c[k]^2, {k, 0, n}]; (* A202856 *)
Table[ch[n], {n, 0, 30}] (* Jean-François Alcover, Aug 10 2018, translated from Maple *)

Formula

a(n) = Sum_{k=0..n} binomial(n, k)*2^(n-k)*c(k)^2, where c() = A126120().

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 ]

a(n) ~ 2^(n-1) * 3^(n+3) / (Pi * n^3). - Vaclav Kotesovec, Jul 20 2019