A001877 -id:A001877 - cp's OEIS frontend

A001879 a(n) = (2n+2)!/(n!*2^(n+1)).

1, 6, 45, 420, 4725, 62370, 945945, 16216200, 310134825, 6547290750, 151242416325, 3794809718700, 102776096548125, 2988412653476250, 92854250304440625, 3070380543400170000, 107655217802968460625, 3989575718580595893750, 155815096120119939628125

Offset: 0

N. J. A. Sloane

From Wolfdieter Lang, Oct 06 2008: (Start)
a(n) is the denominator of the n-th approximant to the continued fraction 1^2/(6+3^2/(6+5^2/(6+... for Pi-3. W. Lang, Oct 06 2008, after an e-mail from R. Rosenthal. Cf. A142970 for the corresponding numerators.
The e.g.f. g(x)=(1+x)/(1-2*x)^(5/2) satisfies (1-4*x^2)*g''(x) - 2*(8*x+3)*g'(x) -9*g(x) = 0 (from the three term recurrence given below). Also g(x)=hypergeom([2,3/2],[1],2*x). (End)
Number of descents in all fixed-point-free involutions of {1,2,...,2(n+1)}. A descent of a permutation p is a position i such that p(i) > p(i+1). Example: a(1)=6 because the fixed-point-free involutions 2143, 3412, and 4321 have 2, 1, and 3 descents, respectively. - Emeric Deutsch, Jun 05 2009
First differences of A193651. - Vladimir Reshetnikov, Apr 25 2016
a(n-2) is the number of maximal elements in the absolute order of the Coxeter group of type D_n. - Jose Bastidas, Nov 01 2021

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77 (Problem 10, values of Bessel polynomials).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Reinis Cirpons, James East, and James D. Mitchell, Transformation representations of diagram monoids, arXiv:2411.14693 [math.RA], 2024. See pp. 3, 33.
Selden Crary, Richard Diehl Martinez and Michael Saunders, The Nu Class of Low-Degree-Truncated Rational Multifunctions. Ib. Integrals of Matern-correlation functions for all odd-half-integer class parameters, arXiv:1707.00705 [stat.ME], 2017, Table 1.
Alexander Kreinin, Integer Sequences and Laplace Continued Fraction, Preprint 2016.
J. Riordan, Notes to N. J. A. Sloane, Jul. 1968

Cf. A002544, A001814, A001876, A001877, A001878.
Second column of triangle A001497. Equals (A001147(n+1)-A001147(n))/2.
Equals row sums of A163938.

[Factorial(2*n+2)/(Factorial(n)*2^(n+1)): n in [0..20]]; // Vincenzo Librandi, Nov 22 2011

restart: G(x):=(1-x)/(1-2*x)^(1/2): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od:x:=0:seq(f[n],n=2..20); # Zerinvary Lajos, Apr 04 2009

Table[(2n+2)!/(n!2^(n+1)),{n,0,20}] (* Vincenzo Librandi, Nov 22 2011 *)

PARI
```
a(n)=if(n<0,0,(2*n+2)!/n!/2^(n+1))
```

E.g.f.: (1+x)/(1-2*x)^(5/2).
a(n)*n = a(n-1)*(2n+1)*(n+1); a(n) = a(n-1)*(2n+4)-a(n-2)*(2n-1), if n>0. - Michael Somos, Feb 25 2004
From Wolfdieter Lang, Oct 06 2008: (Start)
a(n) = (n+1)*(2*n+1)!! with the double factorials (2*n+1)!!=A001147(n+1).
D-finite with recurrence a(n) = 6*a(n-1) + ((2*n-1)^2)*a(n-2), a(-1)=0, a(0)=1. (End)
With interpolated 0's, e.g.f.: B(A(x)) where B(x)= x exp(x) and A(x)=x^2/2.
E.g.f.: -G(0)/2 where G(k) = 1 - (2*k+3)/(1 - x/(x - (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 06 2012
G.f.: (1-x)/(2*x^2*Q(0)) - 1/(2*x^2), where Q(k) = 1 - x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 20 2013
From Karol A. Penson, Jul 12 2013: (Start)
Integral representation as n-th moment of a signed function w(x) of bounded variation on (0,infinity),
  w(x) = -(1/4)*sqrt(2)*sqrt(x)*(1-x)*exp(-x/2)/sqrt(Pi):
  a(n) = Integral_{x>=0} x^n*w(x), n>=0.
  For x>1, w(x)>0. w(0)=w(1)=limit(w(x),x=infinity)=0. For x<1, w(x)<0.
Asymptotics: a(n)->(1/576)*2^(1/2+n)*(1152*n^2+1680*n+505)*exp(-n)*(n)^(n), for n->infinity. (End)
G.f.: 2F0(3/2,2;;2x). - R. J. Mathar, Aug 08 2015

Entry revised Aug 31 2004 (thanks to Ralf Stephan and Michael Somos)

E.g.f. in comment line corrected by Wolfdieter Lang, Nov 21 2011

A001876 Number of divisors of n of the form 5k+1; a(0)=0.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 2, 2, 1, 1, 3, 1, 1, 1, 1, 2, 3, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 1, 4, 1, 1, 1, 1, 2, 3, 1, 1, 1, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 1

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

For numbers of divisors of n of the form 5k+i (i=1, 2, 3, 4) see: this sequence, A001877, A001878, A001899.

Cf. A001620, A256779.

CoefficientList[ Series[ Together[ Sum[ x^n/(1 - x^(5n)), {n, 110}]], {x, 0, 110}], x] (* Robert G. Wilson v, Jan 31 2011 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 5] == 1 &]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Nov 25 2023 *)

a(n) = if(n==0,0, sumdiv(n, d, (d % 5) == 1)); \\ Michel Marcus, Feb 25 2021

G.f.: Sum_{n>=0} x^(5n+1)/(1-x^(5n+1)).
G.f.: Sum_{n>=1} x^n/(1-x^(5*n)). - Joerg Arndt, Jan 30 2011
Sum_{k=1..n} a(k) = n*log(n)/5 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,5) - (1 - gamma)/5 = A256779 - (1 - A001620)/5 = 0.651363... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A001878 Number of divisors of n of the form 5k+3; a(0) = 0.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 2, 0, 0, 1, 0, 1, 2, 0, 1, 1, 1, 0, 1, 0, 1, 2, 0, 0, 2, 0, 1, 2, 1, 0, 1, 1, 0, 1, 1, 0, 3, 0, 0, 1, 1, 1, 2, 0, 2, 1, 1, 0, 1, 0, 0, 2, 1, 1, 2, 0, 1, 2, 0, 0, 3, 1, 0, 1, 1, 0, 3, 0, 1, 1, 0, 1, 2, 0, 1, 1

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A001876, A001877, A001899.

Cf. A001620, A256848.

Join[{0}, Table[d = Divisors[n]; Length[Select[d, Mod[#, 5] == 3 &]], {n, 100}]] (* T. D. Noe, Aug 10 2012 *)
Table[Count[Divisors[n],?(Mod[#,5]==3&)],{n,0,90}] (* _Harvey P. Dale, Nov 08 2012 *)

a(n) = if (n==0, 0, sumdiv(n, d, (d % 5)==3)); \\ Michel Marcus, Feb 28 2021

G.f.: Sum_{n>=0} x^(5*n+3)/(1 - x^(5*n+3)).
G.f.: Sum_{k>=1} x^(3*k)/(1 - x^(5*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/5 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,5) - (1 - gamma)/5 = A256848 - (1 - A001620)/5 = -0.0983206... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A001899 Number of divisors of n of the form 5k+4; a(0) = 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 2, 0, 0, 1, 2, 1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 1, 0, 2, 1, 0, 0, 2, 1, 0, 0, 1, 0, 2, 0, 2, 1, 1, 1, 1, 0, 0, 1, 2, 0, 0, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 1, 1, 1, 1, 0, 0, 3, 0, 0

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A001876, A001877, A001878.

Cf. A001620, A256849.

Join[{0}, Table[d = Divisors[n]; Length[Select[d, Mod[#, 5] == 4 &]], {n, 100}]] (* T. D. Noe, Aug 10 2012 *)

a(n) = if (n==0, 0, sumdiv(n, d, (d % 5)==4)); \\ Michel Marcus, Feb 28 2021

G.f.: Sum_{n>=0} x^(5*n+4)/(1 - x^(5*n+4)).
G.f.: Sum_{k>=1} x^(4*k)/(1 - x^(5*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/5 + c*n + O(n^(1/3)*log(n)), where c = gamma(4,5) - (1 - gamma)/5 = A256849 - (1 - A001620)/5 = -0.213442... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

Better definition from Michael Somos, Aug 31 2004

A363795 Number of divisors of n of the form 7*k + 2.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 0, 2, 0, 1, 0, 2, 1, 1, 0, 1, 0, 1, 0, 2, 1, 2, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 3, 0, 2, 0, 1, 0, 1, 1, 2, 1, 1, 0, 1, 0, 2, 0, 2, 0, 3, 0, 2, 1, 1, 0, 2, 0, 1, 1

Offset: 1

Seiichi Manyama, Jun 23 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A279061, A363805, A363806, A363807, A363808.
Cf. A001822, A001877.
Cf. A284443.
Cf. A001620, A016630, A354628 (psi(2/7)).

a[n_] := DivisorSum[n, 1 &, Mod[#, 7] == 2 &]; Array[a, 100] (* Amiram Eldar, Jun 23 2023 *)

PARI
```
a(n) = sumdiv(n, d, d%7==2);
```

G.f.: Sum_{k>0} x^(2*k)/(1 - x^(7*k)).
G.f.: Sum_{k>0} x^(7*k-5)/(1 - x^(7*k-5)).
Sum_{k=1..n} a(k) = n*log(n)/7 + c*n + O(n^(1/3)*log(n)), where c = gamma(2,7) - (1 - gamma)/7 = 0.188117..., gamma(2,7) = -(psi(2/7) + log(7))/7 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A359237 Number of divisors of 5n-3 of form 5k+1.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 4, 1, 2, 2, 2, 1, 3, 1, 2, 1, 3, 2, 4, 1, 2, 1, 2, 2, 3, 1, 2, 1, 3, 1, 5, 1, 2, 1, 3, 1, 3, 2, 2, 2, 2, 1, 4, 1, 3, 1, 2, 1, 3, 1, 4, 3, 2, 1, 4, 1, 2, 1, 3, 1, 3, 2, 2, 1, 2, 2, 5, 1, 3, 1

Offset: 1

Seiichi Manyama, Dec 22 2022

Also number of divisors of 5*n-3 of form 5*k+2.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A001876, A001877, A359233, A359236, A359238.

a[n_] := DivisorSum[5*n-3, 1 &, Mod[#, 5] == 1 &]; Array[a, 100] (* Amiram Eldar, Aug 23 2023 *)

PARI
```
a(n) = sumdiv(5*n-3, d, d%5==1);
```
PARI
```
a(n) = sumdiv(5*n-3, d, d%5==2);
```

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-x^(5*k-3))))

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^(2*k-1)/(1-x^(5*k-4))))

a(n) = A001876(5*n-3) = A001877(5*n-3).
G.f.: Sum_{k>0} x^k/(1 - x^(5*k-3)).
G.f.: Sum_{k>0} x^(2*k-1)/(1 - x^(5*k-4)).

A359244 Number of divisors of 5n-4 of form 5k+2.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 2, 0, 2, 0, 1, 1, 1, 1, 3, 0, 1, 1, 1, 0, 3, 0, 2, 1, 1, 0, 3, 1, 1, 1, 2, 0, 2, 0, 2, 1, 1, 0, 4, 1, 1, 2, 1, 0, 2, 0, 2, 1, 2, 0, 3, 0, 2, 1, 2, 1, 3, 0, 1, 1, 1, 0, 5, 0, 1, 2, 1, 0, 2, 1, 2, 1, 1, 1, 4, 0, 2, 1, 3, 0, 2, 0, 1, 2, 1

Offset: 1

Seiichi Manyama, Dec 22 2022

Also number of divisors of 5*n-4 of form 5*k+3.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A001877, A001878.

a[n_] := DivisorSum[5*n-4, 1 &, Mod[#, 5] == 2 &]; Array[a, 100] (* Amiram Eldar, Aug 16 2023 *)

PARI
```
a(n) = sumdiv(5*n-4, d, d%5==2);
```
PARI
```
a(n) = sumdiv(5*n-4, d, d%5==3);
```

my(N=100, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^(5*k-2)))))

my(N=100, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(3*k-1)/(1-x^(5*k-3)))))

a(n) = A001877(5*n-4) = A001878(5*n-4).
G.f.: Sum_{k>0} x^(2*k)/(1 - x^(5*k-2)).
G.f.: Sum_{k>0} x^(3*k-1)/(1 - x^(5*k-3)).

A359269 Number of divisors of 5n-2 of form 5k+2.

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 1, 2, 0, 1, 0, 2, 0, 2, 0, 3, 0, 1, 0, 2, 1, 1, 0, 2, 1, 1, 0, 4, 0, 1, 0, 2, 0, 2, 1, 2, 0, 1, 0, 3, 0, 3, 1, 2, 0, 1, 0, 2, 1, 1, 0, 4, 0, 1, 0, 4, 0, 1, 1, 2, 1, 1, 1, 3, 0, 1, 0, 2, 0, 4, 0, 2, 0, 1, 0, 4, 1, 2, 1, 2, 0, 1, 0, 4

Offset: 1

Seiichi Manyama, Dec 23 2022

Also number of divisors of 5*n-2 of form 5*k+4.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A001877, A001899, A359236.

a[n_] := DivisorSum[5*n-2, 1 &, Mod[#, 5] == 2 &]; Array[a, 100] (* Amiram Eldar, Aug 16 2023 *)

PARI
```
a(n) = sumdiv(5*n-2, d, d%5==2);
```
PARI
```
a(n) = sumdiv(5*n-2, d, d%5==4);
```

my(N=100, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^(5*k-1)))))

my(N=100, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(4*k-2)/(1-x^(5*k-3)))))

a(n) = A001877(5*n-2) = A001899(5*n-2).
G.f.: Sum_{k>0} x^(2*k)/(1 - x^(5*k-1)).
G.f.: Sum_{k>0} x^(4*k-2)/(1 - x^(5*k-3)).

A363898 Expansion of Sum_{k>0} k * x^(2k) / (1 - x^(5k)).

Original entry on oeis.org

0, 1, 0, 2, 0, 3, 1, 4, 0, 5, 0, 7, 0, 9, 0, 8, 1, 9, 0, 10, 3, 12, 0, 14, 0, 13, 1, 18, 0, 15, 0, 17, 0, 19, 5, 21, 1, 19, 0, 20, 0, 28, 0, 24, 0, 23, 1, 28, 7, 25, 3, 27, 0, 29, 0, 36, 1, 29, 0, 35, 0, 32, 9, 34, 0, 36, 1, 38, 0, 45, 0, 43, 0, 39, 0, 38, 12, 39, 0, 40, 3, 42, 0, 63, 5, 43, 1

Offset: 1

Seiichi Manyama, Jun 27 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A363897, A363899, A363900.

Cf. A001877, A284280.

a[n_] := DivisorSum[n, # &, Mod[n/#, 5] == 2 &]; Array[a, 100] (* Amiram Eldar, Jun 27 2023 *)

PARI
```
a(n) = sumdiv(n, d, (n/d%5==2)*d);
```

a(n) = Sum_{d|n, n/d==2 mod 5} d.

G.f.: Sum_{k>0} x^(5*k-3) / (1 - x^(5*k-3))^2.

A218445 a(n) = Sum_{k>=0} floor(n/(5*k + 2)).

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 4, 5, 5, 6, 6, 8, 8, 10, 10, 11, 12, 13, 13, 14, 15, 17, 17, 19, 19, 20, 21, 23, 23, 24, 24, 26, 26, 28, 29, 31, 32, 33, 33, 34, 34, 37, 37, 39, 39, 40, 41, 43, 44, 45, 46, 48, 48, 50, 50, 52, 53, 54, 54, 56, 56, 58, 59, 61, 61, 63, 64, 66, 66, 68, 68, 71, 71, 73, 73, 74, 76, 77, 77, 78

Offset: 0

Benoit Cloitre, Oct 28 2012

Seiichi Manyama, Table of n, a(n) for n = 0..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Partial sums of A001877.
Cf. A218444, A218446, A218447.
Cf. A001620, A256780.

Table[Sum[Floor[n/(5k+2)],{k,0,n}],{n,0,80}] (* Harvey P. Dale, Dec 08 2022 *)

A218445[n]:=sum(floor(n/(5*k+2)),k,0,n)$
makelist(A218445[n],n,0,80); /* Martin Ettl, Oct 29 2012 */

PARI
```
a(n)=sum(k=0,n\5,(n\(5*k+2)))
```

a(n) = n*log(n)/5 + c*n + O(n^(1/3)*log(n)), where c = gamma(2,5) - (1 - gamma)/5 = A256780 - (1 - A001620)/5 = 0.105832... (Smith and Subbarao, 1981). - Amiram Eldar, Apr 20 2025

cp's OEIS Frontend

A001879 a(n) = (2n+2)!/(n!*2^(n+1)).

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A001876 Number of divisors of n of the form 5k+1; a(0)=0.

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A001878 Number of divisors of n of the form 5k+3; a(0) = 0.

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A001899 Number of divisors of n of the form 5k+4; a(0) = 0.

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A363795 Number of divisors of n of the form 7*k + 2.

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A359237 Number of divisors of 5*n-3 of form 5*k+1.

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A359244 Number of divisors of 5*n-4 of form 5*k+2.

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A359237 Number of divisors of 5n-3 of form 5k+1.

A359244 Number of divisors of 5n-4 of form 5k+2.

A359269 Number of divisors of 5n-2 of form 5k+2.

A363898 Expansion of Sum_{k>0} k * x^(2k) / (1 - x^(5k)).