Search: id:A138351
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%I A138351
%S A138351 1,0,2,1,11,16,95,232,1085,3460,14820,54275,227095,895688,3756688,
%T A138351 15462293,65586405,277342336,1192038266,5136760581,22357937431,
%U A138351 97730561480,430177280197,1901975209706,8454151507801,37734802709796
%N A138351 Central moment sequence of tr(A^2) in USp(4).
%C A138351 If A is a random matrix in the compact group USp(4) (4 X 4 complex matrices 
               which are unitary and symplectic), then a(n)=E[(tr(A^2)+1)^n] is 
               the nth central moment of the trace of A^2, since E[tr(A^2)] = -1 
               (see A138350).
%D A138351 Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials 
               and random matrices", preprint, 2008.
%H A138351 Kiran S. Kedlaya and Andrew V. Sutherland, <a href="http://arXiv.org/
               abs/0803.4462">Hyperelliptic curves, L-polynomials and random matrices</
               a>.
%F A138351 a(n)=(1/2)Integral_{x=0..Pi,y=0..Pi}(2cos(2x)+2cos(2y)+1)^n(2cos(x)-2cos(y))^2(2/
               Pi*sin^2(x))(2/Pi*sin^2(y))dxdy. a(n)=Sum_{i=0..n}binomial(n,i)A138350(i)
%e A138351 a(4) = 11 because E[((tr(A^2)+1)^4] = 11 for a random matrix A in USp(4).
%e A138351 a(4) = 1*A138350(0)+4*A138350(1)+6*A138350(2)+4*A138350(3)+1*A138350(4)
%e A138351 = 1*1 + 4*(-1) + 6*3 + 4*(-6) + 1*20 = 11.
%Y A138351 Cf. A138350.
%Y A138351 Sequence in context: A158354 A055459 A080958 this_sequence A120293 A063624 
               A101851
%Y A138351 Adjacent sequences: A138348 A138349 A138350 this_sequence A138352 A138353 
               A138354
%K A138351 nonn
%O A138351 0,3
%A A138351 Andrew V. Sutherland (drew(AT)math.mit.edu), Mar 16 2008, Mar 31 2008

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