cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

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

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

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.

Programs

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

Formula

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

Views

Author

N. J. A. Sloane

Keywords

Examples

			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 + ...

References

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

Links

Crossrefs

Cf. A046978, A046981, A052277, A007019, A007452, A009775, A092820.
Essentially the same as absolute values of A007415.

Programs

  • Maple
    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
  • Mathematica
    Denominator[CoefficientList[Series[Exp[x]Sin[x],{x,0,30}],x] ] (* Harvey P. Dale, Feb 14 2015 *)
  • PARI
    a(n) = if (n % 4, n!/2^floor(n/2), 1); \\ Michel Marcus, Oct 12 2015

Formula

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

Views

Author

N. J. A. Sloane

Keywords

Examples

			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-...

References

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

Links

Crossrefs

Cf. A046980, A046979, A052277, A007019, A007415, A009775, A092820.
Absolute values are essentially the same as A007452.

Programs

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

Formula

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

Views

Author

N. J. A. Sloane

Keywords

Crossrefs

Absolute values are essentially the same as A046981, where zeros are replaced by ones.
Cf. A052277, A007019, A046979, A007415, A009775, A092820.

Formula

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

A217260 Expansion of e.g.f. 2*arctan(1+x) - Pi/2.

Original entry on oeis.org

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

Views

Author

Vladimir Kruchinin, Mar 17 2013

Keywords

Links

Crossrefs

Cf. A000111 (e.g.f. sec(x)+tan(x)), A009775.

Programs

  • Maple
    seq(2^(1-n/2)*sin(3/4*Pi*n)*(n-1)!, n=1..50); # Robert Israel, Jan 17 2017
  • Mathematica
    Table[2^(1 - n/2)*(n - 1)!*Sin[3*Pi*n/4], {n, 30}] (* Wesley Ivan Hurt, Oct 14 2023 *)
  • Maxima
    a(n):=n!*sum(((-1)^(n+i)*binomial(n-1,2*i-2))/(2*i-1),i,1,(n+1)/2)/2^(n-1);

Formula

E.g.f.: 2*arctan(1+x) - Pi/2.
a(n) = n!*(Sum_{i=1..floor(n+1)/2} ((-1)^(n+i)*binomial(n-1, 2*i-2))/(2*i-1))/2^(n-1).
E.g.f. is the series reversion of sec(x) + tan(x) - 1.
From Robert Israel, Jan 17 2017: (Start)
a(n) = (n-1)*a(n-1) - (n-1)*(n-2)*a(n-2)/2.
a(n) = 2^(1-n/2)*(n-1)!*sin(3*Pi*n/4). (End)

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

Views

Author

Jon Perry, Feb 26 2004

Keywords

Comments

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

Examples

			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)

Links

Crossrefs

Cf. A052277, A007019.
The function appears in several expansions: A009775, A046979, A046981, A007415, A007452.

Programs

  • Magma
    [Factorial(n) / 2^Floor(n/2): n in [0..25]]; // Vincenzo Librandi, May 14 2011
  • Maple
    a:= n-> n!/2^floor(n/2): seq(a(n), n=0..40);
  • Mathematica
    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)
  • Python
    from math import factorial
def A090932(n): return factorial(n)>>(n>>1) # Chai Wah Wu, Jan 18 2023
  • Sage
    @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

Formula

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)

Extensions

Edited by Ralf Stephan, Sep 07 2004

A296979 Expansion of e.g.f. arcsin(log(1 + x)).

Original entry on oeis.org

0, 1, -1, 3, -12, 68, -480, 4144, -42112, 494360, -6581880, 98079696, -1617373296, 29245459176, -575367843960, 12235339942344, -279650131845120, 6836254328079936, -177979145883651648, 4916243253642325056, -143602294106947553280, 4422411460743707222784
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 22 2017

Keywords

Examples

			arcsin(log(1 + x)) = x^1/1! - x^2/2! + 3*x^3/3! - 12*x^4/4! + 68*x^5/5! - 480*x^6/6! + ...

Crossrefs

Cf. A001710, A001818, A003703, A003708, A009024, A009454, A009775, A104150, A189815, A296980, A296981, A296982.

Programs

  • Maple
    a:=series(arcsin(log(1+x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 21; CoefficientList[Series[ArcSin[Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[-I Log[I Log[1 + x] + Sqrt[1 - Log[1 + x]^2]], {x, 0, nmax}], x] Range[0, nmax]!

Formula

a(n) ~ -(-1)^n * n^(n-1) / (exp(1) - 1)^(n - 1/2). - Vaclav Kotesovec, Mar 26 2019

A296980 Expansion of e.g.f. arcsinh(log(1 + x)).

Original entry on oeis.org

0, 1, -1, 1, 0, -2, -30, 446, -3248, 12412, 16020, -211356, -10756944, 284038272, -3556910448, 19122463296, 135073768320, -1286054192304, -108801241372368, 3952903127312016, -65667347037774720, 339816855220730784, 8862271481944986336
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 22 2017

Keywords

Examples

			arcsinh(log(1 + x)) = x^1/1! - x^2/2! + x^3/3! - 2*x^5/5! - 30*x^6/6! + ...

Crossrefs

Cf. A001710, A001818, A003703, A003708, A009024, A009454, A009775, A104150, A296435, A296979, A296981, A296982.

Programs

  • Maple
    a:=series(arcsinh(log(1+x)),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[ArcSinh[Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Log[Log[1 + x] + Sqrt[1 + Log[1 + x]^2]], {x, 0, nmax}], x] Range[0, nmax]!

A296981 Expansion of e.g.f. arctan(log(1 + x)).

Original entry on oeis.org

0, 1, -1, 0, 6, -22, -30, 952, -5656, -9952, 508320, -3874992, -20690208, 833780400, -7697940432, -52230156288, 2467649024640, -24686997151104, -329724479772288, 14493628861307136, -159114034671287040, -2682505451050592256, 126421889770129637376
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 22 2017

Keywords

Examples

			arctan(log(1 + x)) = x^1/1! - x^2/2! + 6*x^4/4! - 22*x^5/5! - 30*x^6/6! + ...

Crossrefs

Cf. A001710, A003703, A003708, A009024, A009454, A009775, A010050, A104150, A110708, A296979, A296980, A296982.

Programs

  • Maple
    a:=series(arctan(log(1+x)),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[ArcTan[Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[(I/2) Log[1 - I Log[1 + x]] - (I/2) Log[1 + I Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!

Formula

a(n) ~ (-1)^(n+1) * (n-1)! * sin(n*(Pi-1)/2) / (2 - 2*cos(1))^(n/2). - Vaclav Kotesovec, Mar 26 2019

A296982 Expansion of e.g.f. arctanh(log(1 + x)).

Original entry on oeis.org

0, 1, -1, 4, -18, 118, -930, 8888, -98504, 1248784, -17790480, 281590032, -4901447232, 93064850448, -1914144990576, 42396742460928, -1006101059149440, 25466710774651776, -684902462140798848, 19503187752732408576, -586221766070655432960
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 22 2017

Keywords

Examples

			arctanh(log(1 + x)) = x^1/1! - x^2/2! + 4*x^3/3! - 18*x^4/4! + 118*x^5/5! - 930*x^6/6! + ...

Crossrefs

Cf. A001710, A003703, A003708, A009024, A009454, A009775, A010050, A104150, A202139, A296979, A296980, A296981.

Programs

  • Maple
    a:=series(arctanh(log(1+x)),x=0,21): seq(n!*coeff(a,x,n),n=0..20); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 20; CoefficientList[Series[ArcTanh[Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Log[1 + Log[1 + x]]/2 - Log[1 - Log[1 + x]]/2, {x, 0, nmax}], x] Range[0, nmax]!
Showing 1-10 of 12 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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