This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A138354 #17 Nov 29 2024 17:38:31
%S A138354 1,0,3,1,21,26,215,498,2821,9040,43695,165375,752785,3101970,13881803,
%T A138354 59837183,267860685,1184749704,5337504263,23996776941,108964583121,
%U A138354 495544446410,2267450194443,10402298479276,47926692348121
%N A138354 Central moment sequence of tr(A^4) in USp(4).
%C A138354 Binomial transform of A018224.
%C A138354 If A is a random matrix in the compact group USp(4) (4 X 4 complex are unitary and symplectic), then a(n)=E[(tr(A^4)+1)^n] is the n-th central moment of the trace of A^4, since E[tr(A^4)] = -1 (see A018224).
%H A138354 Kiran S. Kedlaya and Andrew V. Sutherland, <a href="http://arXiv.org/abs/0803.4462">Hyperelliptic curves, L-polynomials and random matrices</a>, arXiv:0803.4462 [math.NT], 2008-2010.
%F A138354 a(n) = (1/2)*Integral_{x=0..Pi,y=0..Pi}(2cos(4x)+2cos(4y)+1)^n*(2cos(x)-2cos(y))^2*(2/Pi*sin^2(x))*(2/Pi*sin^2(y))dxdy.
%F A138354 a(n) = Sum_{i=0..n} (-1)^i binomial(n,i)*A018224(i). [corrected by _Jean-François Alcover_, Aug 13 2018]
%F A138354 a(n) = (5*A201805(n) - A201805(n+1))/4. - _Mark van Hoeij_, Nov 29 2024
%e A138354 a(3) = 1 because E[(tr(A^4)+1)^3] = 1.
%e A138354 a(3) = 1*A018224(0) + 3*A018224(1) + 3*A018224(2) + 1*A018224(1) = 1*1 + 3*(-1) + 3*4 + 1*(-9) = 1.
%t A138354 a18224[n_] := Binomial[n, Floor[n/2]]^2;
%t A138354 a[n_] := Sum[(-1)^i Binomial[n, i] a18224[i], {i, 0, n}];
%t A138354 Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Aug 13 2018 *)
%Y A138354 Cf. A018224.
%K A138354 nonn
%O A138354 0,3
%A A138354 _Andrew V. Sutherland_, Mar 16 2008, Mar 31 2008