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A262961 Crandall numbers: (2/Pi)^4 Integral_{t>=0} ([Pi I_0(t)]^2 - [K_0(t)]^2) I_0(t) [K_0(t)]^5 (2t)^(2n-1) dt.

%I A262961 #125 Jun 23 2021 09:06:13
%S A262961 0,1,2,15,302,12559,900288,98986140,15459635718,3251842717671,
%T A262961 885987204390450,303482789415233775,127643176985672421000,
%U A262961 64668997044706349592900,38844990446097247188562800,27296481783843922533011100000,22184577644604207037479874293750
%N A262961 Crandall numbers: (2/Pi)^4 Integral_{t>=0} ([Pi I_0(t)]^2 - [K_0(t)]^2) I_0(t) [K_0(t)]^5 (2t)^(2n-1) dt.
%C A262961 Anton Mellit and David Broadhurst define the sequence to be the "round" of the integral, with the conjecture that this rounding is exact. No one seems to know how to prove that any of the integrals gives a rational number, let alone an integer.
%C A262961 a(0) is not defined: the integral diverges.
%C A262961 Several papers written by Jon Borwein with various coauthors, motivated by work of David Broadhurst, provide recurrence relations for moments of Bessel functions. - _M. F. Hasler_, Oct 11 2015
%C A262961 Named after the American physicist, mathematician and computer scientist Richard Eugene Crandall (1947-2012). - _Amiram Eldar_, Jun 23 2021
%H A262961 M. F. Hasler, <a href="/A262961/b262961.txt">Table of n, a(n) for n = 1..60</a>; first 49 terms from D. Broadhurst. See also the extended table of 450 terms in the Broadhurst link below.
%H A262961 David H. Bailey, Jonathan M. Borwein, David Broadhurst, M. L. Glasser, <a href="https://arxiv.org/abs/0801.0891">Elliptic integral evaluations of Bessel moments</a>, arXiv:0801.0891 [hep-th], 2008.
%H A262961 David H. Bailey, Jonathan M. Borwein, David Broadhurst, M. L. Glasser, <a href="http://dx.doi.org/10.1088/1751-8113/41/20/205203">Elliptic integral evaluations of Bessel moments</a>, J. Phys. A: Math. Theor., Vol. 41 (2008) 205203.
%H A262961 Jonathan M. Borwein, Bruno Salvy, <a href="https://projecteuclid.org/euclid.em/1227118973">A Proof of a Recursion for Bessel Moments</a>, Experiment. Math., Vol. 17, No. 2 (2008), pp. 223-230.
%H A262961 David Broadhurst, <a href="http://physics.open.ac.uk/~dbroadhu/recmem.pdf">Crandall Memorial Puzzle</a>, Oct 04, 2015.
%H A262961 David Broadhurst, <a href="/A262961/a262961.pdf">Crandall Memorial Puzzle</a>. [Cached copy, with permission]
%H A262961 David Broadhurst, <a href="http://physics.open.ac.uk/~dbroadhu/recsol.pdf">Crandall memorial puzzle: solution and heuristics</a>.
%H A262961 David Broadhurst, <a href="/A265079/a265079.pdf">Crandall memorial puzzle: solution and heuristics</a>. [Cached copy, with permission]
%H A262961 David Broadhurst, <a href="/A262961/a262961.txt">Table of n, a(n) for n = 1..450</a>.
%H A262961 David Broadhurst, <a href="/A262961/a262961_1.txt">The largest prime (or noncomposite) factor of A262961(n) for n = 1..94</a>.
%H A262961 David Broadhurst, <a href="http://arxiv.org/abs/1604.03057">Feynman integrals, L-series and Kloosterman moments</a>, arXiv:1604.03057 [physics.gen-ph], 2016. See Eq. 147.
%H A262961 Hans Havermann and David Broadhurst, <a href="http://chesswanks.com/num/CrandallNumbersFactored.txt">Crandall Numbers Factored</a>.
%H A262961 Yajun Zhou, <a href="https://arxiv.org/abs/1706.01068">Hilbert Transforms and Sum Rules of Bessel Moments</a>, arXiv:1706.01068 [math.CA], 2017.
%H A262961 Yajun Zhou, <a href="https://arxiv.org/abs/1801.05555">Some algebraic and arithmetic properties of Feynman diagrams</a>, arXiv:1801.05555 [math.NT], 2018.
%F A262961 a(n) = (2/Pi)^4 Integral_{t>=0} ([Pi I_0(t)]^2 - [K_0(t)]^2) I_0(t) [K_0(t)]^5 (2t)^(2n-1) dt, where I_0(t) and K_0(t) are Bessel functions.
%F A262961 Floor(a(n+1)/a(n)) = A002943(n-2) = 2(n-2)(2n-3) for n > 7; with round() the relation holds for n = 3, ..., 9. - _M. F. Hasler_, Oct 11 2015
%p A262961 ogf := x * BesselI(0,sqrt(x)/2)^4 * BesselK(0,sqrt(x)/2)^4;
%p A262961 S := convert(simplify(asympt(ogf, x, 25)),polynom):
%p A262961 seq(coeff(S,x,-i),i=0..24); # _Mark van Hoeij_, Oct 23 2017
%t A262961 a[n_] := (t1 = NIntegrate[(2*t)^(2*n-1)*BesselI[0, t]^3*BesselK[0, t]^5, {t, 0, Infinity}, WorkingPrecision -> 50]; t2 = NIntegrate[(2*t)^(2*n-1) * BesselI[0, t]*BesselK[0, t]^7, {t, 0, Infinity}, WorkingPrecision -> 50]; Round[(2/Pi)^4*(Pi^2*t1-t2)]); Table[Print["a(", n, ") = ", an = a[n]]; an, {n, 1, 16}] (* _Jean-François Alcover_, Oct 06 2015, adapted from _David Broadhurst_'s PARI script *)
%o A262961 (PARI) { default(realprecision,50); infty=[1]; for(n=1,16, t1=intnum(t=0,[infty,2], besseli(0,t)^3*besselk(0,t)^5*(2*t)^(2*n-1)); t2=intnum(t=0,[infty,6], besseli(0,t)*besselk(0,t)^7*(2*t)^(2*n-1)); print(n," ",round((2/Pi)^4*(t1*Pi^2-t2)))); } /* _David Broadhurst_, Oct 05 2015 */
%o A262961 (PARI) A262961(n,p=max(2*n,20),a=1)={default(realprecision,p); my(i,k,r=1); forprime(q=3,(n-1)\2,r*=q^(2*ceil(n/q)-4)); n=n*2-1; p=Pi^-2; round(intnum(t=0,[[1],a],((i=besseli(0,t))^3*(k=besselk(0,t))^5-i*k^7*p)*t^n)*2^(n+4)/r/Pi^2)*r} \\ It appears that (in PARI V.2.6.1) the parameter a=1 gives much better results for the numerical integration than the "correct" a=2 (resp. a=6 for the second term); combining all in one integral allows evaluation of the Bessel functions and t^(2n-1) only once. - _M. F. Hasler_, Oct 11 2015, improved thanks to a suggestion by _David Broadhurst_, Oct 16 2015
%Y A262961 Cf. A263413 for the largest prime factor of a(n).
%Y A262961 See also A265079.
%K A262961 nonn
%O A262961 1,3
%A A262961 _Andrew R. Reiter_, Oct 05 2015
%E A262961 Offset corrected by _David Broadhurst_, Oct 05 2015