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 A202814 Moments of the quadratic coefficient of the characteristic polynomial of a random matrix in U(1) X U(1) (embedded in USp(4)). 14
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Nachum Dershowitz, Touchardâ€™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 Conjecture: 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 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)]; 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 A131785 A006669 Adjacent sequences:  A202811 A202812 A202813 * A202815 A202816 A202817 KEYWORD nonn AUTHOR N. J. A. Sloane, Dec 24 2011 STATUS approved

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Last modified August 18 00:56 EDT 2019. Contains 326059 sequences. (Running on oeis4.)