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 A228229 Recurrence a(n) = n*(n + 1)*a(n-1) + 1 with a(0) = 1. 4
 1, 3, 19, 229, 4581, 137431, 5772103, 323237769, 23273119369, 2094580743211, 230403881753211, 30413312391423853, 4744476733062121069, 863494765417306034559, 181333900737634267257391, 43520136177032224141773841, 11837477040152764966562484753 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Main subdiagonal (and main superdiagonal) of A099597. Cf. A006040 and A228230. LINKS E. W. Weisstein, Modified Bessel Function of the First Kind FORMULA a(n) = n!*(n + 1)!*sum {k = 0..n} 1/(k!*(k + 1)!). Generating function: 1/(1 - x)*1/sqrt(x)*BesselI(1, 2*sqrt(x)) = sum {n >= 0} a(n)*x^n/(n!*(n + 1)!). Defining recurrence equation: a(n) = n*(n + 1)*a(n-1) + 1 with a(0) = 1. Alternative recurrence equation: a(0) = 1, a(1) = 3, and for n >= 2, a(n) = (n*(n + 1) + 1)*a(n-1) - n*(n - 1)*a(n-2). The sequence b(n) := n!*(n + 1)! satisfies the same recurrence with the initial conditions b(0) = 1, b(1) = 2. It follows that we have the finite continued fraction expansion a(n) = n!*(n + 1)!*(1/(1 - 1/(3 - 2/(7 - 6/(13 - … - n*(n - 1)/(n^2 + n + 1)))))). Taking the limit yields the continued fraction expansion for the modified Bessel function value BesselI(1,2) = sum {k = 0..inf} 1/(k!*(k + 1)!) = 1/(1 - 1/(3 - 2/(7 - 6/(13 - ...- n*(n - 1)/(n^2 + n + 1 - ...))))) = 1.590636... (see A096789). a(n) ~ BesselI(1,2) * n!*(n+1)!. - Vaclav Kotesovec, May 06 2015 MAPLE A228229 :=proc(n) option remember     if n = 0 then 1     else n*(n+1)*procname(n-1) + 1     end if: end proc: seq(A228229(n), n = 0..20); MATHEMATICA RecurrenceTable[{a[n] == n*(n + 1)*a[n-1] + 1, a[0] == 1}, a, {n, 0, 20}] (* Vaclav Kotesovec, May 06 2015 *) CROSSREFS Cf. A006040, A096789, A099597, A228230. Sequence in context: A126444 A198046 A295812 * A001929 A230316 A157675 Adjacent sequences:  A228226 A228227 A228228 * A228230 A228231 A228232 KEYWORD nonn,easy AUTHOR Peter Bala, Aug 19 2013 STATUS approved

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Last modified November 20 06:06 EST 2018. Contains 317385 sequences. (Running on oeis4.)