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A226975
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Decimal expansion I_1(1), the modified Bessel function of the first kind.
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0
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5, 6, 5, 1, 5, 9, 1, 0, 3, 9, 9, 2, 4, 8, 5, 0, 2, 7, 2, 0, 7, 6, 9, 6, 0, 2, 7, 6, 0, 9, 8, 6, 3, 3, 0, 7, 3, 2, 8, 8, 9, 9, 6, 2, 1, 6, 2, 1, 0, 9, 2, 0, 0, 9, 4, 8, 0, 2, 9, 4, 4, 8, 9, 4, 7, 9, 2, 5, 5, 6, 4, 0, 9, 6, 4, 3, 7, 1, 1, 3, 4, 0, 9, 2, 6, 6, 4, 9, 9, 7, 7, 6, 6, 8, 1, 4, 4, 1, 0, 0, 6, 4, 6, 7, 7, 8, 8, 6
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OFFSET
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0,1
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COMMENTS
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This is also the derivative of the zeroth modified Bessel function at 1.
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LINKS
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FORMULA
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I_1(1) = (1/2) * Sum_{k>=0} (2*k)/(4^k*k!^2) = (1/2) * Sum_{k>=0} (2*k)/A002454(k).
Equals (1/2) * Sum_{k>=0} (4*k^2 + 4*k - 1) / (2*k)!!^2.
Equals exp(-1) * Sum_{k>=0} binomial(2*k,k+1)/(2^k*k!).
Equals (-e) * Sum_{k>=0} (-1/2)^k * binomial(2*k,k+1)/k!
Equals (1/Pi)*Integral_{t=0..Pi} exp(cos(t))*cos(t) dt. (End)
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EXAMPLE
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0.56515910399248502720769602760986330732889962162109...
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MATHEMATICA
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RealDigits[BesselI[1, 1], 10, 110][[1]]
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PROG
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(SageMath)
((1/2) * sum(1 / (4^x * factorial(x) * rising_factorial(2, x)), x, 0, oo)).n(360)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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