A211374 -id:A211374 - cp's OEIS frontend

A141310 The odd numbers interlaced with the constant-2 sequence.

1, 2, 3, 2, 5, 2, 7, 2, 9, 2, 11, 2, 13, 2, 15, 2, 17, 2, 19, 2, 21, 2, 23, 2, 25, 2, 27, 2, 29, 2, 31, 2, 33, 2, 35, 2, 37, 2, 39, 2, 41, 2, 43, 2, 45, 2, 47, 2, 49, 2, 51, 2, 53, 2, 55, 2, 57, 2, 59, 2, 61, 2, 63, 2, 65, 2, 67, 2, 69, 2, 71, 2, 73, 2, 75, 2, 77, 2, 79, 2, 81, 2, 83, 2, 85, 2, 87, 2, 89, 2, 91, 2, 93, 2, 95, 2, 97

Offset: 0

Paul Curtz, Aug 02 2008

Similarly, the principle of interlacing a sequence and its first differences leads from A000012 and its differences A000004 to A059841, or from A140811 and its first differences A017593 to a sequence -1, 6, 5, 18, ...
If n is even then a(n) = n + 1 ; otherwise a(n) = 2. - Wesley Ivan Hurt, Jun 05 2013
Denominators of floor((n+1)/2) / (n+1), n > 0. - Wesley Ivan Hurt, Jun 14 2013
a(n) is also the number of minimum total dominating sets in the (n+1)-gear graph for n>1. - Eric W. Weisstein, Apr 11 2018
a(n) is also the number of minimum total dominating sets in the (n+1)-sun graph for n>1. - Eric W. Weisstein, Sep 09 2021
Denominators of Cesàro means sequence of A114112, corresponding numerators are in A354008. -  Bernard Schott, May 14 2022
Also, denominators of Cesàro means sequence of A237420, corresponding numerators are in A354280. -  Bernard Schott, May 22 2022

Antti Karttunen, Table of n, a(n) for n = 0..16384
Eric Weisstein's World of Mathematics, Gear Graph.
Eric Weisstein's World of Mathematics, Minimum Total Dominating Set.
Eric Weisstein's World of Mathematics, Sun Graph.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A000004, A000012, A000142, A005408, A008586, A017593, A059841, A060747, A109613, A140811, A211374, A319702 (rgs-transform).
Cf. A114112, A354008.
Cf. A237420, A354280.

a:= n-> n+1-(n-1)*(n mod 2): seq(a(n), n=0..96); # Wesley Ivan Hurt, Jun 05 2013

Flatten[Table[{2 n - 1, 2}, {n, 40}]] (* Alonso del Arte, Jun 15 2013 *)
Riffle[Range[1, 79, 2], 2] (* Alonso del Arte, Jun 14 2013 *)
Table[((-1)^n (n - 1) + n + 3)/2, {n, 0, 20}] (* Eric W. Weisstein, Apr 11 2018 *)
Table[Floor[(n + 1)/2]/(n + 1), {n, 0, 20}] // Denominator (* Eric W. Weisstein, Apr 11 2018 *)
LinearRecurrence[{0, 2, 0, -1}, {2, 3, 2, 5}, {0, 20}] (* Eric W. Weisstein, Apr 11 2018 *)
CoefficientList[Series[(1 + 2 x + x^2 - 2 x^3)/(-1 + x^2)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Apr 11 2018 *)

A141310(n) = if(n%2,2,1+n); \\ (for offset=0 version) - Antti Karttunen, Oct 02 2018

A141310off1(n) = if(n%2,n,2); \\ (for offset=1 version) - Antti Karttunen, Oct 02 2018

def A141310(n): return 2 if n % 2 else n + 1 # Chai Wah Wu, May 24 2022

a(2n) = A005408(n). a(2n+1) = 2.
First differences: a(n+1) - a(n) = (-1)^(n+1)*A109613(n-1), n > 0.
b(2n) = -A008586(n), and b(2n+1) = A060747(n), where b(n) = a(n+1) - 2*a(n).
a(n) = 2*a(n-2) - a(n-4). - R. J. Mathar, Feb 23 2009
G.f.: (1+2*x+x^2-2*x^3)/((x-1)^2*(1+x)^2). - R. J. Mathar, Feb 23 2009
From Wesley Ivan Hurt, Jun 05 2013: (Start)
a(n) = n + 1 - (n - 1)*(n mod 2).
a(n) = (n + 1) * (n - floor((n+1)/2))! / floor((n+1)/2)!.
a(n) = A000142(n+1) / A211374(n+1). (End)

Edited by R. J. Mathar, Feb 23 2009

Term a(45) corrected, and more terms added by Antti Karttunen, Oct 02 2018

A249163 Triangle read by rows: the positive terms of A163626.

Original entry on oeis.org

1, 1, 1, 2, 1, 12, 1, 50, 24, 1, 180, 360, 1, 602, 3360, 720, 1, 1932, 25200, 20160, 1, 6050, 166824, 332640, 40320, 1, 18660, 1020600, 4233600, 1814400, 1, 57002, 5921520, 46070640, 46569600, 3628800, 1, 173052, 33105600, 451725120, 898128000, 239500800

Offset: 0

Paul Curtz, Dec 15 2014

We have two possibilities: with or without 0's.
Without 0's:
1,
1,
1,   2,
1,  12,
1,  50,  24,
1, 180, 360,
etc.
Sum of every row: A000670(n).
First two terms of successive columns: 1, 1, 2, 12, 24, 360, ... = A211374.
With 0's:
1,   0,    0,   0,
1,   0,    0,   0,
1,   2,    0,   0,
1,  12,    0,   0,
1,  50,   24,   0,
1, 180,  360,   0,
1, 602, 3360, 720,
etc.
The columns are essentially A000012, A028243, A028246, A228909, A228911, A228913, from Stirling numbers of the second kind S(n,3), S(n,5), S(n,7), S(n,9), S(n,11), ... .

Cf. A163626, A000670, A211374; also A000012, A000392, A000481, A000771, A049447, A028243, A028246, A091137, A228909, A163626, A228911, A228913 and Worpitzky numbers for the second Bernoulli numbers A164555(n)/A027642(n).

Derivative[0][y][x] = y[x]; Derivative[1][y][x] = y[x]*(1 - y[x]); Derivative[n_][y][x] := Derivative[n][y][x] = D[Derivative[n - 1][y][x], x]; row[n_] := CoefficientList[Derivative[n][y][x], y[x]] // Rest; Table[ Select[row[n], Positive] , {n, 0, 12}] // Flatten
(* or, simply: *) Table[(-1)^k*k!*StirlingS2[n+1, k+1], {n, 0, 12}, {k, 0, n}] // Flatten // Select[#, Positive]& (* Jean-François Alcover, Dec 16 2014 *)

A226731 a(n) = (2n - 1)!/(2n).

Original entry on oeis.org

20, 630, 36288, 3326400, 444787200, 81729648000, 19760412672000, 6082255020441600, 2322315553259520000, 1077167364120207360000, 596585001666576384000000, 388888194657798291456000000

Offset: 3

Wesley Ivan Hurt, Jun 15 2013

For n < 3, the formula does not produce an integer.

The ratio of the product of the partition parts of 2n into exactly two parts to the sum of the partition parts of 2n into exactly two parts. For example, a(3) = 20, and 2*3 = 6 has 3 partitions into exactly two parts: (5,1), (4,2), (3,3). Forming the ratio of product to sum (of parts), we have (5*1*4*2*3*3)/(5+1+4+2+3+3) = 360/18 = 20. - Wesley Ivan Hurt, Jun 24 2013

			a(3) = (2*3 - 1)!/(2*3) = 5!/6 = 120/6 = 20.

Seiichi Manyama, Table of n, a(n) for n = 3..225
Index entries for sequences related to partitions.

Cf. A001105, A002674, A005843, A009445, A093353, A143623, A211374.

seq((2*k-1)!/(2*k),k=1..20); # Wesley Ivan Hurt, Jun 24 2013

Table[(2n-1)!/(2n),{n,3,20}] (* Harvey P. Dale, Jun 19 2013 *)

a(n) = (2*n-1)!/(2*n); \\ Michel Marcus, Nov 01 2016

a(n) = A009445(n-1)/A005843(n) = A002674(n)/A001105(n). - Wesley Ivan Hurt, Jun 24 2013
a(n) ~ sqrt(Pi)*2^(2*n-1)*n^(2*n-3/2)/exp(2*n). - Ilya Gutkovskiy, Nov 01 2016
From Amiram Eldar, Mar 10 2022: (Start)
Sum_{n>=3} 1/a(n) = e - 8/3.
Sum_{n>=3} (-1)^(n+1)/a(n) = cos(1) + sin(1) - 4/3. (End)

A248812 Repeated terms of (2n)! (A010050).

Original entry on oeis.org

1, 1, 2, 2, 24, 24, 720, 720, 40320, 40320, 3628800, 3628800, 479001600, 479001600, 87178291200, 87178291200, 20922789888000, 20922789888000, 6402373705728000, 6402373705728000, 2432902008176640000, 2432902008176640000, 1124000727777607680000

Offset: 0

Wesley Ivan Hurt, Oct 16 2014

For n>1, a(n) is the product of the smallest parts in the partitions of 4*floor(n/2) = A168273(n) into two parts.

Index entries for sequences related to partitions

Cf. A000142, A052928, A010050, A168273, A211374.

[Factorial(2*Floor(n/2)) : n in [0..20]];

A248812:=n->(2*floor(n/2))!: seq(A248812(n), n=0..20);

Mathematica
```
Table[(2*Floor[n/2])!, {n, 0, 20}]
```

a(n) = ( 2*floor(n/2) )! = A000142(A052928(n)).
a(2n) = a(2n+1) = A010050(n) = A211374(2n-1).
E.g.f.: log((1+x)/(1-x))/2+1/(1-x^2). - Robert Israel, Oct 19 2014

cp's OEIS Frontend

A141310 The odd numbers interlaced with the constant-2 sequence.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

A249163 Triangle read by rows: the positive terms of A163626.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A226731 a(n) = (2n - 1)!/(2n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A248812 Repeated terms of (2n)! (A010050).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula