A052277 -id:A052277 - cp's OEIS frontend

A009775 Exponential generating function is tanh(log(1+x)).

0, 1, -1, 0, 6, -30, 90, 0, -2520, 22680, -113400, 0, 7484400, -97297200, 681080400, 0, -81729648000, 1389404016000, -12504636144000, 0, 2375880867360000, -49893498214560000, 548828480360160000, 0, -151476660579404160000

Offset: 0

R. H. Hardin

Brandon Humpert and Jeremy L. Martin, The Incidence Hopf Algebra of Graphs, DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), pp. 517-526. [See Example 3.4]

Cf. A052277, A007019, A046979, A007415, A007452, A092820, A217260.

A009775(n)=polcoeff(tanh(log(1+x+O(x^n)*x)),n)*n!  \\ M. F. Hasler, Oct 10 2012

a(0) = 0, a(4n+3) = 0, a(n) = (-1)^[n == 2, 5, 8 mod 8] * n!/2^floor(n/2). - Ralf Stephan, Mar 06 2004
From Peter Bala, Nov 25 2011: (Start)
(1): a(n) = i*n!/2^(n+1)*{(i-1)^(n+1)-(-1-i)^(n+1)} for n>=1.
The function tanh(log(1+x)) is a disguised form of the rational function (x^2+2*x)/(x^2+2*x+2). Observe that
(2): (x^2+2*x)/(x^2+2*x+2) = d/dx[x - atan((x^2+2*x)/(2*x+2))].
Hence, with an offset of 1, the egf for this sequence is
(3): x - atan((x^2+2*x)/(2*x+2)) = x^2/2! - x^3/3! + 6*x^5/5!- 30*x^6/6! + 90*x^7/7! - ....
This sequence is closely related to the series reversion of the function E(x)-1, where E(x) = sec(x)+tan(x) is the egf for the sequence of zigzag numbers A000111. Under the change of variable x -> sec(x)+tan(x)-1 the rational function (x^2+2*x)/(2*x+2) transforms to tan(x). Hence atan((x^2+2*x)/(2*x+2)) is the inverse function of sec(x)+tan(x)-1.
Recurrence relation:
(4): 2*a(n)+2*n*a(n-1)+n*(n-1)*a(n-2) = 0 with a(1) = 1, a(2) = -1.
(End)

Extended with signs by Olivier Gérard, Mar 15 1997

A007415 Expand sin x / exp x = x-x^2+x^3/3-x^5/30+... and invert nonzero coefficients.

Original entry on oeis.org

0, 1, -1, 3, 0, -30, 90, -630, 0, 22680, -113400, 1247400, 0, -97297200, 681080400, -10216206000, 0, 1389404016000, -12504636144000, 237588086736000, 0, -49893498214560000, 548828480360160000, -12623055048283680000, 0, 3786916514485104000000

Offset: 0

N. J. A. Sloane

sign

J. Roberts, Lure of the Integers, Annotated scanned copy of pp. 81, 86 with notes.

Absolute values are essentially the same as A046979, where zeros are replaced by ones.
a(4n+2) = -(-1)^n*A052277(n), a(2n+1) = (-1)^[n/2]*A007019(n).
Cf. A046981, A007452, A009775, A092820.

a:= n-> (p-> `if`(p=0,0,1/p))(coeff(series(sin(x)/exp(x), x, n+1), x, n)):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 17 2017

1/CoefficientList[Sin[x]/Exp[x] + O[x]^26, x] /. ComplexInfinity -> 0 // Quiet (* Jean-François Alcover, Feb 26 2019 *)

a(n) = [n mod 4 > 0] * (-1)^(n+1+[n/4]) * n!/2^[n/2]. - Ralf Stephan, Mar 06 2004

E.g.f.: sin(x)/exp(x) = x-x^2/(G(0)+x); G(k)=2k+1-x+x*(2k+1)/(4k+3-x+x^2*(4k+3)/( (2k+2)*(4k+5)-x^2+x*(2k+2)*(4k+5)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 20 2011

A046979 Denominators of Taylor series for exp(x)*sin(x).

Original entry on oeis.org

1, 1, 1, 3, 1, 30, 90, 630, 1, 22680, 113400, 1247400, 1, 97297200, 681080400, 10216206000, 1, 1389404016000, 12504636144000, 237588086736000, 1, 49893498214560000, 548828480360160000, 12623055048283680000, 1, 3786916514485104000000, 49229914688306352000000

Offset: 0

N. J. A. Sloane

			1*x + 1*x^2 + 1/3*x^3 - 1/30*x^5 - 1/90*x^6 - 1/630*x^7 + 1/22680*x^9 + 1/113400*x^10 + ...

G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.

T. D. Noe, Table of n, a(n) for n=0..100
Herbert S. Wilf, Generatingfunctionology, Academic Press, NY, 1994. See p. 54.

Cf. A046978, A046981, A052277, A007019, A007452, A009775, A092820.

Essentially the same as absolute values of A007415.

a:= n-> denom(coeff(series(sin(x)/exp(x), x, n+1), x, n)):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 17 2017

Denominator[CoefficientList[Series[Exp[x]Sin[x],{x,0,30}],x] ] (* Harvey P. Dale, Feb 14 2015 *)

a(n) = if (n % 4, n!/2^floor(n/2), 1); \\ Michel Marcus, Oct 12 2015

a(4n) = 1, a(n) = n!/2^floor(n/2).

A046981 Denominators of Taylor series for exp(x)*cos(x).

Original entry on oeis.org

1, 1, 1, 3, 6, 30, 1, 630, 2520, 22680, 1, 1247400, 7484400, 97297200, 1, 10216206000, 81729648000, 1389404016000, 1, 237588086736000, 2375880867360000, 49893498214560000, 1, 12623055048283680000, 151476660579404160000, 3786916514485104000000

Offset: 0

N. J. A. Sloane

			1+1*x-1/3*x^3-1/6*x^4-1/30*x^5+1/630*x^7+1/2520*x^8+1/22680*x^9-...

G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.

T. D. Noe, Table of n, a(n) for n=0..100

Cf. A046980, A046979, A052277, A007019, A007415, A009775, A092820.

Absolute values are essentially the same as A007452.

Denominator/@CoefficientList[Series[Exp[x]Cos[x],{x,0,30}],x] (* Harvey P. Dale, Jun 28 2011 *)

a(4n+2) = 1, a(n) = n!/2^[n/2].

A007452 Expand cos x / exp x and invert nonzero coefficients.

Original entry on oeis.org

1, -1, 0, 3, -6, 30, 0, -630, 2520, -22680, 0, 1247400, -7484400, 97297200, 0, -10216206000, 81729648000, -1389404016000, 0, 237588086736000, -2375880867360000, 49893498214560000, 0, -12623055048283680000

Offset: 0

N. J. A. Sloane

sign

Absolute values are essentially the same as A046981, where zeros are replaced by ones.

Cf. A052277, A007019, A046979, A007415, A009775, A092820.

a(4n+2) = 0, a(n) = (-1)^[n == 1, 4, 7 mod 8] * n!/2^floor(n/2). - Ralf Stephan, Mar 06 2004

A090932 a(n) = n! / 2^floor(n/2).

Original entry on oeis.org

1, 1, 1, 3, 6, 30, 90, 630, 2520, 22680, 113400, 1247400, 7484400, 97297200, 681080400, 10216206000, 81729648000, 1389404016000, 12504636144000, 237588086736000, 2375880867360000, 49893498214560000, 548828480360160000, 12623055048283680000, 151476660579404160000

Offset: 0

Jon Perry, Feb 26 2004

nonn

Number of permutations of the n-th row of Pascal's triangle.
Can be seen as the multiplicative equivalent to the generalized pentagonal numbers. - Peter Luschny, Oct 13 2012
a(n) is the number of permutations of [n] in which all ascents start at an even position. For example, a(3) = 3 counts 213, 312, 321. - David Callan, Nov 25 2021

			From _Rigoberto Florez_, Apr 07 2017: (Start)
a(5) = 5!/2^2 = 120/4 = 30.
a(6) = 6!/2^3 = 1*6*15 = 90.
a(7) = 7!/2^3 = 3*10*21 = 630. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Rigoberto Flórez and Leandro Junes, A relation between triangular numbers and prime numbers, Integers, Vol. 12, No. 1 (2012), pp. 83-96.

Cf. A052277, A007019.

The function appears in several expansions: A009775, A046979, A046981, A007415, A007452.

[Factorial(n) / 2^Floor(n/2): n in [0..25]]; // Vincenzo Librandi, May 14 2011

a:= n-> n!/2^floor(n/2): seq(a(n), n=0..40);

Table[n!/2^Floor[n/2], {n, 0, 21}] (* Michael De Vlieger, Jul 25 2016 *)
nxt[{n_,a_,b_}]:={n+1,b,a Binomial[n,2]}; NestList[nxt,{2,1,1},30][[All,2]] (* Harvey P. Dale, Aug 26 2022 *)

PARI
```
a(n)=n!/2^floor(n/2)
```

from math import factorial
def A090932(n): return factorial(n)>>(n>>1) # Chai Wah Wu, Jan 18 2023

@CachedFunction
def A090932(n):
    if n == 0 : return 1
    fact = n//2 if is_even(n) else n
    return fact * A090932(n-1)
[A090932(n) for n in (0..21)] # Peter Luschny, Oct 13 2012

a(n) = binomial(n-1, 2) * a(n-2).
E.g.f.: (1+x)/(1-1/2*x^2).
E.g.f.: G(0)  where G(k) = 1 + x/(1 - x/(x + 2/G(k+1) )) ; (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 27 2012
G.f.: G(0), where G(k)= 1 + (2*k+1)*x/(1 - x*(k+1)/(x*(k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 28 2013
a(n) = (n+1)!/A093968(n+1). - Anton Zakharov, Jul 25 2016
a(n) ~ sqrt(2*Pi*n)*exp(-n)*n^n/2^floor(n/2). - Ilya Gutkovskiy, Jul 25 2016
From Rigoberto Florez, Apr 07 2017: (Start)
if n=2k, n! / 2^k = t(1)t(3)t(5)...t(2k-1),
if n=2k+1, n! / 2^k = t(2)t(4)t(6)...t(2k),
if n=2k, n! / 2^k = (t(k)-t(0))*(t(k)-t(1))*...*(t(k)-t(k-1)),
with t(i)= i-th triangular number. (End)
From Amiram Eldar, Feb 25 2022: (Start)
Sum_{n>=0} 1/a(n) = cosh(sqrt(2)) + sinh(sqrt(2))/sqrt(2).
Sum_{n>=0} (-1)^n/a(n) = cosh(sqrt(2)) - sinh(sqrt(2))/sqrt(2). (End)

Edited by Ralf Stephan, Sep 07 2004

A052278 a(n) = (4n+3)!/4^n.

Original entry on oeis.org

6, 1260, 2494800, 20432412000, 475176173472000, 25246110096567360000, 2658415393168543008000000, 501882242076289234480320000000, 157671325170687025904337331200000000, 77811744999685071405945898971187200000000, 57616484702466807973246700352205274112000000000

Offset: 0

N. J. A. Sloane, Feb 05 2000

nonn

Cf. A052277.

Table[(4n+3)!/4^n,{n,0,20}] (* Harvey P. Dale, Mar 14 2020 *)

a(n) = (4*n+3)!/4^n; \\ Michel Marcus, Feb 25 2022

From Amiram Eldar, Feb 25 2022: (Start)
Sum_{n>=0} 1/a(n) = (sinh(sqrt(2)) - sin(sqrt(2)))/(4*sqrt(2)).
Sum_{n>=0} (-1)^n/a(n) = (cosh(1)*sin(1) - sinh(1)*cos(1))/4. (End)

cp's OEIS Frontend

A009775 Exponential generating function is tanh(log(1+x)).

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A007415 Expand sin x / exp x = x-x^2+x^3/3-x^5/30+... and invert nonzero coefficients.

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Maple

Mathematica

Formula

A046979 Denominators of Taylor series for exp(x)*sin(x).

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Author

Keywords

Examples

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Programs

Maple

Mathematica

PARI

Formula

A046981 Denominators of Taylor series for exp(x)*cos(x).

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Examples

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Programs

Mathematica

Formula

A007452 Expand cos x / exp x and invert nonzero coefficients.

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A090932 a(n) = n! / 2^floor(n/2).

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A052278 a(n) = (4n+3)!/4^n.

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