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A087547
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a(n) = n!*2^(n+1) * (Integral_{x = 0..1} 1/(1+x^2)^(n+1) dx) - Pi*(2*n)!/(2^(n+1)*n!).
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7
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0, 1, 4, 22, 160, 1464, 16224, 211632, 3179520, 54092160, 1028113920, 21594021120, 496702402560, 12418039065600, 335293281792000, 9723592350259200, 301432670532403200, 9947299050359193600, 348155822449999872000, 12881771833023700992000, 502389223133024747520000
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OFFSET
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0,3
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COMMENTS
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a(n)/A001147 gives an approximation for Pi/2 with (n-1)/3 + 1 digits of accuracy. - Aaron Kastel, Nov 13 2012
a(n) is the number of linear chord diagrams on 2n vertices with one marked chord such that none of the remaining n-1 chords are excluded by (i.e., are outside and do not contain) the marked chord, see [Young]. - Donovan Young, Aug 11 2020
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LINKS
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FORMULA
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a(n) = (2*n)!/(n!*2^n)*(Sum_{k = 0..n-1} 2^k*k!^2/(2*k+1)!). Thus a(n)/ ((2*n)!/(n!*2^n)) -> Pi/2 as n -> infinity since Sum_{k >= 0} 2^k*k!^2/(2*k+1)! = Pi/2.
It appears that a(n) = Sum_{k = 1..n} 2^(k-1)*(k-1)!*(n+k-1)!/(2*k-1)!. Cf. A167571.
a(n) = (2*n)!/(n!*2^n)*(Pi/2) - 2^(n+1)*n!*(Integral_{x = 0..1} x^(2*n)/(1 + x^2)^(n+1) dx). Cf. A068102. (End)
Recurrence equation: a(n) = (3*n - 2)*a(n-1) - (n - 1)*(2*n - 3)*a(n-2) with a(1) = 1 and a(2) = 4.
The sequence b(n) = A001147(n), beginning [1, 3, 15, 105, 945, ... ], satisfies the same second-order recurrence equation. This leads to the generalized continued fraction expansion lim_{n -> infinity} a(n)/b(n) = Pi/2 = 1 + 1/(3 - 6/(7 - 15/(10 - ... - n*(2*n - 1)/((3*n + 1) - ... )))). (End)
E.g.f.: arctan(x/sqrt(1 - 2*x))/sqrt(1 - 2*x). - Donovan Young, Aug 11 2020
a(n) = (n - 1)!*Sum_{k=0..n - 1} binomial(2*n - 1, 2*k)/binomial(n - 1, k).
a(n)/(n-1)! = 1 + (2*n - 1)/2*Integral_{t = 0..Pi/2} (1 + sin(2*t))^(n - 1) - (1 - sin(2*t))^(n - 1)dt.
Bala's conjecture is true. (End)
a(n) = (n - 1)!/2 * Sum_{k=0..n - 1} binomial(2*n, 2*k + 1)/binomial(n - 1, k). - Sela Fried, May 18 2023
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EXAMPLE
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a(3) = 22.
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MAPLE
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f := proc(n) 4*n!*2^(n-1) * (int (1/(1+x^2)^(n+1), x=0..1)) - Pi*(2*n)!/(2^(n+1)*n!); end; # N. J. A. Sloane, Oct 30 2003
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MATHEMATICA
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f[n_] := Simplify[n!*2^(n + 1)*(Integrate[ 1/(1 + x^2)^(n + 1), {x, 0, 1}]) - Pi(2n)!/(2^(n + 1)*n!)]; Table[ f[n], {n, 0, 20}] (* Robert G. Wilson v, Oct 31 2003 *)
CoefficientList[Normal[Series[1/Sqrt[1-2*x]*ArcTan[x/Sqrt[1-2*x]], {x, 0, 10}]]/.{x^n_.->x^n*n!}, x] (* Donovan Young, Aug 11 2020 *)
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PROG
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(Magma) [0] cat [n eq 1 select 1 else (2*n-1)*Self(n-1)+Factorial(n-1): n in [1..25]]; // Vincenzo Librandi, Nov 07 2014
(Magma) I:=[1, 4]; [0] cat [n le 2 select I[n] else (3*n-2)*Self(n-1)-(n-1)*(2*n-3)*Self(n-2): n in [1..25] ]; // Vincenzo Librandi, Feb 19 2015
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Al Hakanson (hawkuu(AT)excite.com), Oct 24 2003
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EXTENSIONS
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STATUS
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approved
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