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Search: id:A006040
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| A006040 |
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a(n+1) = n^2 a(n) + 1.
(Formerly M1950)
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+0
5
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| 1, 2, 9, 82, 1313, 32826, 1181737, 57905114, 3705927297, 300180111058, 30018011105801, 3632179343801922, 523033825507476769, 88392716510763573962
(list; graph; listen)
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OFFSET
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1,2
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REFERENCES
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R. K. Guy, personal communication.
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LINKS
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Index entries for sequences related to Bessel functions or polynomials
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FORMULA
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Nearest integer to BesselI(0, 2)*n!*n!, n>2.
a(n) = n!^2*Sum_{k=0..n} 1/k!^2. BesselI(0, 2*sqrt(x))/(1-x) = Sum_{n>=0} a(n)*x^n/n!^2. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 30 2002
Recurrence: a(1) = 1, a(2) = 2, a(n+1) = (n^2+1)*a(n) - (n-1)^2*a(n-1), n >= 2. The sequence b(n) := (n-1)!^2 satisfies the same recurrence with the initial conditions b(1) = 1, b(2) = 1. It follows that a(n) = n!^2*(1 + 1/(1 - 1/(5 - 4/(10 - ...-(n-1)^2/(n^2+1))))). Hence BesselI(0,2) := sum {k = 0..inf} 1/k!^2 = 1 + 1/(1 - 1/(5 - 4/(10 - ...-(n-1)^2/(n^2+1 - ...)))). Cf. A073701. - Peter Bala (pbala(AT)toucansurf.com), Jul 09 2008
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CROSSREFS
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Main diagonal of array A099597.
Sequence in context: A112670 A117581 A123570 this_sequence A067309 A087798 A113146
Cf. A073701.
Adjacent sequences: A006037 A006038 A006039 this_sequence A006041 A006042 A006043
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KEYWORD
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nonn,easy
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AUTHOR
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njas, Simon Plouffe (plouffe(AT)math.uqam.ca)
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