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

0,2

Links

Table of n, a(n) for n=0..28.

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).

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

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 *)

Crossrefs

Cf. A126869, A126120, A202814.

Sequence in context: A257273 A119021 A002890 * A118929 A287252 A204064

Adjacent sequences: A202853 A202854 A202855 * A202857 A202858 A202859

Keyword

nonn

Author

N. J. A. Sloane, Dec 25 2011

Status

approved