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A087547 a(n) = n!*2^(n+1) * (Integral_{x = 0..1} 1/(1+x^2)^(n+1) dx - Pi*(2*n)!/(2^(n+1)*n!). 6
0, 1, 4, 22, 160, 1464, 16224, 211632, 3179520, 54092160, 1028113920, 21594021120, 496702402560, 12418039065600, 335293281792000, 9723592350259200, 301432670532403200, 9947299050359193600, 348155822449999872000, 12881771833023700992000, 502389223133024747520000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n)/A001147 gives an approximation for Pi/2 with (n-1)/3 + 1 digits of accuracy. - Aaron Kastel, Nov 13 2012

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..404

Richard P. Brent, Hideyuki Ohtsuka, Judy-anne H. Osborn, Helmut Prodinger, Some binomial sums involving absolute values, arXiv:1411.1477 [math.CO], 2014 (see page 15).

FORMULA

a(n) = (2n-1)*a(n-1) + (n-1)!. - Aaron Kastel, Nov 13 2012

From Peter Bala, Jun 21 2013: (Start)

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 -> inf since sum {k = 0..inf} 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!*(int {x = 0..1} x^(2*n)/(1+x^2)^(n+1) dx). Cf. A068102. (End)

From Peter Bala, Feb 18 2015: (Start)

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 limit {n -> inf} a(n)/b(n) = Pi/2 = 1 + 1/(3 - 6/(7 - 15/(10 - ... - n*(2*n - 1)/((3*n + 1) - ... )))). (End)

EXAMPLE

a(3) = 22.

MAPLE

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

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A068102, A167571, A001147.

Sequence in context: A113717 A124563 A122704 * A218678 A184942 A000779

Adjacent sequences:  A087544 A087545 A087546 * A087548 A087549 A087550

KEYWORD

nonn,easy

AUTHOR

Al Hakanson (hawkuu(AT)excite.com), Oct 24 2003

EXTENSIONS

More terms from N. J. A. Sloane, Oct 30 2003

STATUS

approved

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Last modified February 21 10:39 EST 2018. Contains 299390 sequences. (Running on oeis4.)