A167571 -id:A167571 - cp's OEIS frontend

A087547 a(n) = n!2^(n+1) (Integral_{x = 0..1} 1/(1+x^2)^(n+1) dx) - Pi(2n)!/(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

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

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

			a(3) = 22.

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.

Cf. A068102, A167571, A001147, A336601.

[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

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

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

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 *)

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

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

A167569 The lower left triangle of the ED2 array A167560.

Original entry on oeis.org

1, 2, 4, 6, 16, 32, 24, 80, 192, 384, 120, 480, 1344, 3072, 6144, 720, 3360, 10752, 27648, 61440, 122880, 5040, 26880, 96768, 276480, 675840, 1474560, 2949120, 40320, 241920, 967680, 3041280, 8110080, 19169280, 41287680, 82575360

Offset: 1

Johannes W. Meijer, Nov 10 2009

We discovered that the numbers that appear in the lower left triangle of the ED2 array A167560 (m <= n) behave in a regular way, see the formula below. This rather simple regularity doesn't show up in the upper right triangle of the ED2 array (m > n).

			The first few triangle rows are:
[1]
[2, 4]
[6, 16, 32]
[24, 80, 192, 384]
[120, 480, 1344, 3072, 6144]
[720, 3360, 10752, 27648, 61440, 122880]

G. C. Greubel, Table of n, a(n) for the first 50 rows

A167560 is the ED2 array.
A047053, 2*A034177 and A167570 are the first three right hand triangle columns.
A000142, 4*A001715, 32*A001725, 384* A049388 and 6144* A049398 are the first five left hand triangle columns.
A167571 equals the row sums.

a := proc(n, m): 4^(m-1)*(m-1)!*(n+m-1)!/(2*m-1)! end: seq(seq(a(n, m), m=1..n), n=1..8); # Johannes W. Meijer, revised Nov 23 2012

Flatten[Table[4^(m - 1)*(m - 1)!*(n + m - 1)!/(2*m - 1)!, {n, 1, 50}, {m, n}]] (* G. C. Greubel, Jun 16 2016 *)

a(n,m) = 4^(m-1)*(m-1)!*(n+m-1)!/(2*m-1)!.

cp's OEIS Frontend

A087547 a(n) = n!2^(n+1) (Integral_{x = 0..1} 1/(1+x^2)^(n+1) dx) - Pi(2n)!/(2^(n+1)*n!).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Magma

Maple

Mathematica

Formula

Extensions

A167569 The lower left triangle of the ED2 array A167560.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

cp's OEIS Frontend

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!).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Magma

Maple

Mathematica

Formula

Extensions

A167569 The lower left triangle of the ED2 array A167560.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A087547 a(n) = n!2^(n+1) (Integral_{x = 0..1} 1/(1+x^2)^(n+1) dx) - Pi(2n)!/(2^(n+1)*n!).