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 1-4 of 4 results.

A214225 E.g.f. satisfies: A(x) = x/(1 - tanh(A(x))).

Original entry on oeis.org

1, 2, 12, 112, 1440, 23616, 471296, 11085824, 300349440, 9211187200, 315448860672, 11932326789120, 494098626904064, 22230301612703744, 1079857012109475840, 56326462301645307904, 3140024293968001892352, 186308007164786201591808, 11722541029509094870876160
Offset: 1

Views

Author

Paul D. Hanna, Jul 07 2012

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! +...
Related expansions:
A(x) = x + x*tanh(x) + d/dx x^2*tanh(x)^2/2! + d^2/dx^2 x^3*tanh(x)^3/3! + d^3/dx^3 x^4*tanh(x)^4/4! +...
log(A(x)/x) = tanh(x) + d/dx x*tanh(x)^2/2! + d^2/dx^2 x^2*tanh(x)^3/3! + d^3/dx^3 x^3*tanh(x)^4/4! +...
A(x)/x = 1 + x + 4*x^2/2! + 28*x^3/3! + 288*x^4/4! + 3936*x^5/5! + 67328*x^6/6! +...+ A201595(n)*x^n/n! +...
tanh(A(x)) = x + 2*x^2/2! + 10*x^3/3! + 88*x^4/4! + 1096*x^5/5! + 17616*x^6/6! + 346704*x^7/7! + 8072576*x^8/8! +...

Links

Crossrefs

Cf. A201595, A202617, A214222, A214223, A214224.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x-x*Tanh[x],{x,0,20}],x],x]*Range[0,20]!] (* Vaclav Kotesovec, Sep 17 2013 *)
Flatten[{1,Table[1/2*Sum[Binomial[n,k]*k^(n-1),{k,0,n}],{n,2,20}]}] (* Vaclav Kotesovec, Sep 17 2013 *)
  • PARI
    {a(n)=(1/2)*sum(k=0,n,binomial(n,k)*k^(n-1))}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {a(n)=(n-1)!*polcoeff(x/(1 - tanh(x+x*O(x^n)))^n,n)}
  • PARI
    {a(n)=n!*polcoeff(serreverse(x-x*tanh(x+x*O(x^n))), n)}
  • PARI
    {a(n)=n!*polcoeff(sum(k=1, n, k^(k-1)*cosh(k*x +x*O(x^n))*x^k/k! +x*O(x^n)), n)} \\ Paul D. Hanna, Nov 20 2012
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^m*tanh(x+x*O(x^n))^m/m!)); n!*polcoeff(A, n)}
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*tanh(x+x*O(x^n))^m/m!)+x*O(x^n))); n!*polcoeff(A, n)}

Formula

E.g.f. A(x) satisfies:
(1) A(x - x*tanh(x)) = x.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*tanh(x)^n/n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*tanh(x)^n/n! ).
(4) A(x) = Sum_{n>=1} n^(n-1) * cosh(n*x) * x^n / n!. - Paul D. Hanna, Nov 20 2012
(5) A(x) = log(G(x)) where G(x) = exp(x*(1+G(x)^2)/2) is the e.g.f. of A202617. - Paul D. Hanna, Nov 20 2012
a(n) = n*A201595(n-1).
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*k^(n-1).
a(n) = (n-1)! * [x^n] x/(1 - tanh(x))^n.
a(n) = A038049(n)/2. - R. J. Mathar, Peter Bala, Mar 24 2013
a(n) ~ 1/2 * n^(n-1) * sqrt((1+LambertW(1/exp(1)))) / (exp(1)*LambertW(1/exp(1)))^n. - Vaclav Kotesovec, Sep 17 2013

A214224 E.g.f. satisfies: A(x) = x/(1 - tan(A(x))).

Original entry on oeis.org

1, 2, 12, 128, 1920, 37056, 874496, 24395776, 785387520, 28658483200, 1168842842112, 52692107132928, 2601710674640896, 139635670319366144, 8094064830515773440, 503939620849307353088, 33539757103898142179328, 2376284247629812872511488, 178564437032337539449487360
Offset: 1

Views

Author

Paul D. Hanna, Jul 07 2012

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 1920*x^5/5! +...
Related expansions:
A(x) = x + x*tan(x) + d/dx x^2*tan(x)^2/2! + d^2/dx^2 x^3*tan(x)^3/3! + d^3/dx^3 x^4*tan(x)^4/4! +...
log(A(x)/x) = tan(x) + d/dx x*tan(x)^2/2! + d^2/dx^2 x^2*tan(x)^3/3! + d^3/dx^3 x^3*tan(x)^4/4! +...
A(x)/x = 1 + x + 4*x^2/2! + 32*x^3/3! + 384*x^4/4! + 6176*x^5/5! + 124928*x^6/6! +...+ A201594(n)*x^n/n! +...
tan(A(x)) = x + 2*x^2/2! + 14*x^3/3! + 152*x^4/4! + 2296*x^5/5! + 44496*x^6/6! + 1052848*x^7/7! + 29425024*x^8/8! +...

Links

Crossrefs

Cf. A201594, A214222, A214223, A214225.

Programs

  • Maple
    f:= b*(1-tan(b))-x:
newt:= unapply(b-normal(f/diff(f,b)),b):
B:= x:
for n from 1 to 5 do
  B:= convert(series(newt(B),x,2^n+1),polynom)
od:
seq(coeff(B,x,j)*j!, j=1..2^5); # Robert Israel, Feb 04 2019
  • Mathematica
    m = 20; CoefficientList[InverseSeries[Series[x(1-Tan[x]), {x, 0, m}], x]/x, x] Range[m]! (* Jean-François Alcover, Apr 01 2019 *)
  • PARI
    {a(n)=(n-1)!*polcoeff(x/(1 - tan(x+x*O(x^n)))^n,n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {a(n)=n!*polcoeff(serreverse(x-x*tan(x+x*O(x^n))), n)}
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^m*tan(x+x*O(x^n))^m/m!)); n!*polcoeff(A, n)}
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*tan(x+x*O(x^n))^m/m!)+x*O(x^n))); n!*polcoeff(A, n)}

Formula

E.g.f. A(x) satisfies:
(1) A(x - x*tan(x)) = x.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*tan(x)^n/n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*tan(x)^n/n! ).
a(n) = n*A201594(n-1).
a(n) = (n-1)! * [x^n] x/(1 - tan(x))^n.

A214222 E.g.f. satisfies: A(x) = x/(1 - sin(A(x))).

Original entry on oeis.org

1, 2, 12, 116, 1560, 26886, 565376, 14036392, 401823360, 13030976650, 472154276352, 18903994333212, 828807273828352, 39491616319733774, 2032038033784995840, 112293378446546611280, 6632975513529162694656, 417050432063319431036178, 27809989478829060358799360
Offset: 1

Views

Author

Paul D. Hanna, Jul 07 2012

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 116*x^4/4! + 1560*x^5/5! +...
Related expansions:
A(x) = x + x*sin(x) + d/dx x^2*sin(x)^2/2! + d^2/dx^2 x^3*sin(x)^3/3! + d^3/dx^3 x^4*sin(x)^4/4! +...
log(A(x)/x) = sin(x) + d/dx x*sin(x)^2/2! + d^2/dx^2 x^2*sin(x)^3/3! + d^3/dx^3 x^3*sin(x)^4/4! +...
A(x)/x = 1 + x + 4*x^2/2! + 29*x^3/3! + 312*x^4/4! + 4481*x^5/5! + 80768*x^6/6! +...+ A201627(n)*x^n/n! +...
sin(A(x)) = x + 2*x^2/2! + 11*x^3/3! + 104*x^4/4! + 1381*x^5/5! + 23616*x^6/6! + 493975*x^7/7! + 12216448*x^8/8! +...

Crossrefs

Cf. A201627, A214223, A214224, A214225.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x - x*Sin[x],{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 12 2014 *)
  • PARI
    {a(n)=(n-1)!*polcoeff(x/(1 - sin(x+x*O(x^n)))^n,n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {a(n)=n!*polcoeff(serreverse(x-x*sin(x+x*O(x^n))), n)}
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^m*sin(x+x*O(x^n))^m/m!)); n!*polcoeff(A, n)}
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*sin(x+x*O(x^n))^m/m!)+x*O(x^n))); n!*polcoeff(A, n)}

Formula

E.g.f. A(x) satisfies:
(1) A(x - x*sin(x)) = x.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*sin(x)^n/n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*sin(x)^n/n! ).
a(n) = n*A201627(n-1).
a(n) = (n-1)! * [x^n] x/(1 - sin(x))^n.
a(n) ~ sqrt((1-t)/(2+t)) * n^(n-1) * (sqrt(1-t^2)/(1-t)^2)^n / exp(n), where t = 0.527766122670442778... is the root of the equation t = sin(sqrt((1-t)/(1+t))). - Vaclav Kotesovec, Jan 12 2014

A201628 E.g.f. satisfies: A(x) = 1/(1 - sinh(x*A(x))).

Original entry on oeis.org

1, 1, 4, 31, 360, 5601, 109568, 2586151, 71555200, 2271961825, 81441188352, 3253620672303, 143361363439616, 6907049546879041, 361245668908466176, 20383791705206338807, 1234336634416972726272, 79843983527411321710401, 5494767253686351671459840, 400863405346004202504321343
Offset: 0

Views

Author

Paul D. Hanna, Dec 03 2011

Keywords

Comments

The function 1/(1-sinh(x)) is the e.g.f. of A006154, where A006154(n) is the number of labeled ordered partitions of an n-set into odd parts.

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 31*x^3/3! + 360*x^4/4! + 5601*x^5/5! +...
The coefficients in initial powers of G(x) = 1/(1 - sinh(x)) begin:
G^1: [(1), 1, 2, 7, 32, 181, 1232, 9787, 88832, ..., A006154(n), ...];
G^2: [1,(2), 6, 26, 144, 962, 7536, 67706, ...];
G^3: [1, 3,(12), 63, 408, 3123, 27552, 275103, ...];
G^4: [1, 4, 20,(124), 920, 7924, 77600, 850924, ...];
G^5: [1, 5, 30, 215,(1800), 17225, 185280, 2211515, ...];
G^6: [1, 6, 42, 342, 3192,(33606), 393792, 5080662, ...];
G^7: [1, 7, 56, 511, 5264, 60487, (766976), 10634911, ...];
G^8: [1, 8, 72, 728, 8208, 102248, 1395072,(20689208), ...]; ...
where coefficients in parenthesis form initial terms of this sequence:
[1/1, 2/2, 12/3, 124/4, 1800/5, 33606/6, 766976/7, 20689208/8, ...].

Crossrefs

Cf. A214223, A201627, A201595, A006154.

Programs

  • PARI
    {a(n)=n!*polcoeff(1/x*serreverse(x*(1-sinh(x+x^2*O(x^n)))),n)}
  • PARI
    {a(n)=n!*polcoeff(1/(1 - sinh(x+x*O(x^n)))^(n+1)/(n+1), n)}

Formula

E.g.f. A(x) satisfies: A( x*(1 - sinh(x)) ) = 1/(1 - sinh(x)).
E.g.f.: (1/x)*Series_Reversion( x*(1 - sinh(x)) ).
a(n) = [x^n] 1/(1 - sinh(x))^(n+1) / (n+1).
a(n) = A214223(n+1)/(n+1).
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.