cp's OEIS Frontend

A202814 Moments of the quadratic coefficient of the characteristic polynomial of a random matrix in U(1) X U(1) (embedded in USp(4)).

%I A202814 #55 Apr 20 2025 03:47:10
%S A202814 1,2,8,32,148,712,3584,18496,97444,521096,2820448,15414016,84917584,
%T A202814 470982176,2627289344,14728751872,82928400164,468699173576,
%U A202814 2657978454944,15118824666496,86230489902928,493021885470496,2825114755879424,16221295513400576,93312601350167824,537693975424462112,3103220029717015424
%N A202814 Moments of the quadratic coefficient of the characteristic polynomial of a random matrix in U(1) X U(1) (embedded in USp(4)).
%H A202814 G. C. Greubel, <a href="/A202814/b202814.txt">Table of n, a(n) for n = 0..1000</a>
%H A202814 Nachum Dershowitz, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Dershowitz/dersh3.html">Touchard's Drunkard</a>, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.
%H A202814 Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, <a href="http://arxiv.org/abs/1110.6638">Sato-Tate distributions and Galois endomorphism modules in genus 2</a> arXiv:1110.6638 [math.NT], 2011 (the sequence b-hat(n) defined at the end of Section 5.1.1).
%F A202814 a(n) = Sum_{k=0..n} binomial(n, k)*2^(n-k)*b(k)^2, where b() = A126869().
%F A202814 E.g.f.: (exp(x) * I_0(2*x))^2 = square of e.g.f. of A002426. - _Michael Somos_, Jun 27 2012
%F A202814 From _Mark van Hoeij_, May 07 2013: (Start)
%F A202814 a(n) is the constant term of (2+x+y+1/x+1/y)^n.
%F A202814 G.f.: hypergeom([1/2, 1/2],[1],16*x^2/(1-2*x)^2)/(1-2*x). (End)
%F A202814 G.f.: 1 / AGM(1-6*x, 1+2*x), where AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean. - _Paul D. Hanna_, Aug 31 2014
%F A202814 D-finite with recurrence n^2*a(n) +2*(-3*n^2+3*n-1)*a(n-1) -4*(n-1)^2*a(n-2) +24*(n-1) *(n-2)*a(n-3)=0. - _R. J. Mathar_, Jun 14 2016
%F A202814 a(n) ~ 2^(n-1) * 3^(n+1) / (Pi*n). - _Vaclav Kotesovec_, Jul 20 2019
%F A202814 From _Peter Bala_, May 30 2024: (Start)
%F A202814 a(n) = Sum_{k = 0..n} binomial(n, 2*k) * binomial(2*k, k)^2 * 2^(n-2*k).
%F A202814 a(n) = 2^n * hypergeom([-n/2, (-n+1)/2, 1/2], [1, 1], 4). (End)
%e A202814 1 + 2*x + 8*x^2 + 32*x^3 + 148*x^4 + 712*x^5 + 3584*x^6 + 18496*x^7 + 97444*x^8 + ...
%p A202814 b:=n->coeff((x^2+1)^n,x,n); #A126869
%p A202814 bh:=n->add(binomial(n,k)*2^(n-k)*b(k)^2,k=0..n);
%p A202814 [seq(bh(n),n=0..30)];
%p A202814 # alternative program (faster for large n)
%p A202814 seq(simplify(2^n * hypergeom([-n/2, (-n+1)/2, 1/2], [1, 1], 4)), n = 0..30); #  _Peter Bala_, May 30 2024
%t A202814 a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ (Exp[x] BesselI[0, 2 x])^2, {x, 0, n}]] (* _Michael Somos_, Jun 27 2012 *)
%o A202814 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); n! * polcoeff( (exp(x + A) * besseli( 0, 2*x + A))^2, n))} /* _Michael Somos_, Jun 27 2012 */
%o A202814 (PARI) {a(n)=polcoeff( 1 / agm(1-6*x, 1+2*x +x*O(x^n)), n)}
%o A202814 for(n=0, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Aug 31 2014
%Y A202814 Cf. A126869 (the sequence b(n)), A202856.
%Y A202814 Cf. A002426.
%K A202814 nonn,easy
%O A202814 0,2
%A A202814 _N. J. A. Sloane_, Dec 24 2011