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A262961
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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.
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3
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0, 1, 2, 15, 302, 12559, 900288, 98986140, 15459635718, 3251842717671, 885987204390450, 303482789415233775, 127643176985672421000, 64668997044706349592900, 38844990446097247188562800, 27296481783843922533011100000, 22184577644604207037479874293750
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OFFSET
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1,3
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COMMENTS
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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.
a(0) is not defined: the integral diverges.
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
Named after the American physicist, mathematician and computer scientist Richard Eugene Crandall (1947-2012). - Amiram Eldar, Jun 23 2021
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LINKS
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M. F. Hasler, Table of n, a(n) for n = 1..60; first 49 terms from D. Broadhurst. See also the extended table of 450 terms in the Broadhurst link below.
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FORMULA
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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.
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
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MAPLE
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ogf := x * BesselI(0, sqrt(x)/2)^4 * BesselK(0, sqrt(x)/2)^4;
S := convert(simplify(asympt(ogf, x, 25)), polynom):
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MATHEMATICA
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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 *)
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PROG
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(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 */
(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
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CROSSREFS
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Cf. A263413 for the largest prime factor of a(n).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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