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

0,2

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).

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)

Example

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

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 *)

Prog

(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

Crossrefs

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

Cf. A002426.

Sequence in context: A129400 A003304 A150864 * A054116 A348905 A131785

Adjacent sequences: A202811 A202812 A202813 * A202815 A202816 A202817

Keyword

nonn,easy

Author

N. J. A. Sloane, Dec 24 2011

Status

approved