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!).

0, 1, 4, 22, 160, 1464, 16224, 211632, 3179520, 54092160, 1028113920, 21594021120, 496702402560, 12418039065600, 335293281792000, 9723592350259200, 301432670532403200, 9947299050359193600, 348155822449999872000, 12881771833023700992000, 502389223133024747520000

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

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

Links

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

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

Sela Fried and Toufik Mansour, Graph labelings obtainable by random walks, arXiv:2304.05728 [math.CO], 2023.

Sela Fried and Toufik Mansour, Further results on random walk labelings, arXiv:2305.09971 [math.CO], 2023.

Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.

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 -> 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)

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 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

From Sela Fried, Apr 13 2023: (Start)

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) ~ Pi * 2^(n - 1/2) * n^n / exp(n). - Vaclav Kotesovec, Apr 13 2023

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

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, Oct 30 2003

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 *)
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 *)

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, A336601.

Sequence in context: A113717 A124563 A122704 * A218678 A346968 A184942

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