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A202856 Moments of the quadratic coefficient of the characteristic polynomial of a random matrix in SU(2) X SU(2) (inside USp(4)). 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 10 13:30 EDT 2020. Contains 336381 sequences. (Running on oeis4.)