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

1, 2, 8, 32, 148, 712, 3584, 18496, 97444, 521096, 2820448, 15414016, 84917584, 470982176, 2627289344, 14728751872, 82928400164, 468699173576, 2657978454944, 15118824666496, 86230489902928, 493021885470496, 2825114755879424, 16221295513400576, 93312601350167824, 537693975424462112, 3103220029717015424

Offset: 0

Author

N. J. A. Sloane, Dec 24 2011

Keywords

• nonn
• easy

Examples

			1 + 2*x + 8*x^2 + 32*x^3 + 148*x^4 + 712*x^5 + 3584*x^6 + 18496*x^7 + 97444*x^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.
• Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2 arXiv:1110.6638 [math.NT], 2011 (the sequence b-hat(n) defined at the end of Section 5.1.1).

Crossrefs

Cf. A126869 (the sequence b(n)), A202856.

Cf. A002426.

Programs

Maple

b:=n->coeff((x^2+1)^n,x,n); #A126869
bh:=n->add(binomial(n,k)*2^(n-k)*b(k)^2,k=0..n);
[seq(bh(n),n=0..30)];
# alternative program (faster for large n)
seq(simplify(2^n * hypergeom([-n/2, (-n+1)/2, 1/2], [1, 1], 4)), n = 0..30); #  Peter Bala, May 30 2024

Mathematica

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ (Exp[x] BesselI[0, 2 x])^2, {x, 0, n}]] (* Michael Somos, Jun 27 2012 *)

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 */

PARI

{a(n)=polcoeff( 1 / agm(1-6*x, 1+2*x +x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 31 2014

Formula

a(n) = Sum_{k=0..n} binomial(n, k)*2^(n-k)*b(k)^2, where b() = A126869().

E.g.f.: (exp(x) * I_0(2*x))^2 = square of e.g.f. of A002426. - Michael Somos, Jun 27 2012

From Mark van Hoeij, May 07 2013: (Start)

a(n) is the constant term of (2+x+y+1/x+1/y)^n.

G.f.: hypergeom([1/2, 1/2],[1],16*x^2/(1-2*x)^2)/(1-2*x). (End)

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

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

a(n) ~ 2^(n-1) * 3^(n+1) / (Pi*n). - Vaclav Kotesovec, Jul 20 2019

From Peter Bala, May 30 2024: (Start)

a(n) = Sum_{k = 0..n} binomial(n, 2*k) * binomial(2*k, k)^2 * 2^(n-2*k).

a(n) = 2^n * hypergeom([-n/2, (-n+1)/2, 1/2], [1, 1], 4). (End)