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A138349
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Moment sequence of tr(A) in USp(4).
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3
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1, 0, 1, 0, 3, 0, 14, 0, 84, 0, 594, 0, 4719, 0, 40898, 0, 379236, 0, 3711916, 0, 37975756, 0, 403127256, 0, 4415203280, 0, 49671036900, 0, 571947380775, 0, 6721316278650, 0, 80419959684900, 0, 977737404590100, 0, 12058761323277900, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| An aerated version of A005700, which is the main entry for this sequence.
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))^n] is the n-th moment of the trace of A.
The multiplicity of the trivial representation in the n-th tensor power of the standard representation of USp(4).
Number of returning NESW walks of length n on a 2-d integer lattice remaining in the chamber x>=y>=0, same as A005700(n/2) for n even.
Under a generalized Sato-Tate conjecture, this is the moment sequence of the distribution of scaled Frobenius traces a_p/sqrt(p) (as p varies), for almost all genus 2 curves. - Andrew V. Sutherland (drew(AT)math.mit.edu), Mar 16 2008
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REFERENCES
| David J. Grabiner and Peter Magyar, "Random walks in Weyl chambers and the decomposition of tensor powers", Journal of Algebraic Combinatorics, vol. 2 (1993), no. 3, pp 239-260.
Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials and random matrices", preprint, 2008.
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LINKS
| Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices.
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FORMULA
| a(n)=(1/2)Integral_{x=0..Pi,y=0..Pi}(2cos(x)+2cos(y))^n(2cos(x)-2cos(y))^2(2/Pi*sin^2(x))(2/Pi*sin^2(y))dxdy. a(n)=A126120(n)A126120(n+4)-A126120(n+2)^2. a(2n)=A005700(n)=A000108(n)A000108(n+2)-A000108(n+1)^2, a(2n+1)=0.
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EXAMPLE
| a(4)=3 because E[(tr(A)^4] = 3 for a random matrix A in USp(4).
a(4)=3 because A126120(4)A126120(8)-A126120(6)^2 = 2*14-5*5 = 3.
a(4)=3 because EEWW, EWEW and ENSW are the returning walks on Z^2 with x>=y>=0.
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CROSSREFS
| Cf. A005700, A126120, A000108.
Sequence in context: A057374 A058896 A008403 * A135399 A065121 A167339
Adjacent sequences: A138346 A138347 A138348 * A138350 A138351 A138352
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KEYWORD
| easy,nonn
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AUTHOR
| Andrew V. Sutherland (drew(AT)math.mit.edu), Mar 16 2008
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