A214222 -id:A214222 - cp's OEIS frontend

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

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

Offset: 1

Paul D. Hanna, Jul 07 2012

nonn

			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! +...

G. C. Greubel, Table of n, a(n) for n = 1..370

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

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 *)

{a(n)=(1/2)*sum(k=0,n,binomial(n,k)*k^(n-1))}
for(n=1, 25, print1(a(n), ", "))

{a(n)=(n-1)!*polcoeff(x/(1 - tanh(x+x*O(x^n)))^n,n)}

{a(n)=n!*polcoeff(serreverse(x-x*tanh(x+x*O(x^n))), n)}

{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), ", "))

{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)}

{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)}

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

A201627 E.g.f. satisfies: A(x) = 1/(1 - sin(x*A(x))).

Original entry on oeis.org

1, 1, 4, 29, 312, 4481, 80768, 1754549, 44647040, 1303097665, 42923116032, 1575332861101, 63754405679104, 2820829737123841, 135469202252333056, 7018336152909163205, 390175030207597805568, 23169468447962190613121, 1463683656780476860989440, 98016257612539018485477821

Offset: 0

Paul D. Hanna, Dec 03 2011

nonn

Coefficients in the expansion of 1/(1-sin(x)) yield the Euler numbers (A000111).

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 29*x^3/3! + 312*x^4/4! + 4481*x^5/5! +...
The coefficients in initial powers of G(x) = 1/(1 - sin(x)) begin:
G^1: [(1), 1, 2, 5, 16, 61, 272, 1385, 7936, ..., A000111(n+1), ...];
G^2: [1,(2), 6, 22, 96, 482, 2736, 17302, ...];
G^3: [1, 3,(12), 57, 312, 1923, 13152, 98697, ...];
G^4: [1, 4, 20,(116), 760, 5524, 44000, 380516, ...];
G^5: [1, 5, 30, 205,(1560), 13025, 118080, 1153105, ...];
G^6: [1, 6, 42, 330, 2856,(26886), 272832, 2963850, ...];
G^7: [1, 7, 56, 497, 4816, 50407, (565376), 6754097, ...];
G^8: [1, 8, 72, 712, 7632, 87848, 1078272,(14036392), ...]; ...
where coefficients in parenthesis form initial terms of this sequence:
[1/1, 2/2, 12/3, 116/4, 1560/5, 26886/6, 565376/7, 14036392/8, ...].

Cf. A214222, A201594, A000111.

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

{a(n)=n!*polcoeff(1/(1-sin(x+x*O(x^n)))^(n+1)/(n+1), n)}

E.g.f. A(x) satisfies: A( x*(1 - sin(x)) ) = 1/(1 - sin(x)).
E.g.f.: (1/x)*Series_Reversion( x*(1 - sin(x)) ).
a(n) = [x^n] 1/(1 - sin(x))^(n+1) / (n+1).
a(n) = A214222(n+1)/(n+1).

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

Paul D. Hanna, Jul 07 2012

nonn

			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! +...

Robert Israel, Table of n, a(n) for n = 1..362

Cf. A201594, A214222, A214223, A214225.

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

m = 20; CoefficientList[InverseSeries[Series[x(1-Tan[x]), {x, 0, m}], x]/x, x] Range[m]! (* Jean-François Alcover, Apr 01 2019 *)

{a(n)=(n-1)!*polcoeff(x/(1 - tan(x+x*O(x^n)))^n,n)}
for(n=1, 25, print1(a(n), ", "))

{a(n)=n!*polcoeff(serreverse(x-x*tan(x+x*O(x^n))), n)}

{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)}

{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)}

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.

A214223 E.g.f. satisfies: A(x) = x/(1 - sinh(A(x))).

Original entry on oeis.org

1, 2, 12, 124, 1800, 33606, 766976, 20689208, 643996800, 22719618250, 895853071872, 39043448067636, 1863697724715008, 96698693656306574, 5418685033626992640, 326140667283301420912, 20983722785088536346624, 1437191703493403790787218, 104400577820040681757736960

Offset: 1

Paul D. Hanna, Jul 07 2012

nonn

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 124*x^4/4! + 1800*x^5/5! +...
Related expansions:
A(x) = x + x*sinh(x) + d/dx x^2*sinh(x)^2/2! + d^2/dx^2 x^3*sinh(x)^3/3! + d^3/dx^3 x^4*sinh(x)^4/4! +...
log(A(x)/x) = sinh(x) + d/dx x*sinh(x)^2/2! + d^2/dx^2 x^2*sinh(x)^3/3! + d^3/dx^3 x^3*sinh(x)^4/4! +...
A(x)/x = 1 + x + 4*x^2/2! + 31*x^3/3! + 360*x^4/4! + 5601*x^5/5! + 109568*x^6/6! +...+ A201628(n)*x^n/n! +...
sinh(A(x)) = x + 2*x^2/2! + 13*x^3/3! + 136*x^4/4! + 1981*x^5/5! + 37056*x^6/6! + 846777*x^7/7! + 22861952*x^8/8! +...

Cf. A201628, A214222, A214224, A214225.

Rest[CoefficientList[InverseSeries[Series[x - x*Sinh[x],{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 12 2014 *)

{a(n)=(n-1)!*polcoeff(x/(1 - sinh(x+x*O(x^n)))^n,n)}
for(n=1, 25, print1(a(n), ", "))

{a(n)=n!*polcoeff(serreverse(x-x*sinh(x+x*O(x^n))), n)}

{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*sinh(x+x*O(x^n))^m/m!)); n!*polcoeff(A, n)}

{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)*sinh(x+x*O(x^n))^m/m!)+x*O(x^n))); n!*polcoeff(A, n)}

E.g.f. A(x) satisfies:
(1) A(x - x*sinh(x)) = x.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*sinh(x)^n/n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*sinh(x)^n/n! ).
a(n) = n*A201628(n-1).
a(n) = (n-1)! * [x^n] x/(1 - sinh(x))^n.
a(n) ~ n^(n-1) / (sqrt(s+(2-s^2)*cosh(s)) * exp(n) * (s^2*cosh(s))^(n-1/2)), where s = 0.465767175470891411756875... is the root of the equation s*cosh(s) = 1-sinh(s). - Vaclav Kotesovec, Jan 12 2014

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

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A201627 E.g.f. satisfies: A(x) = 1/(1 - sin(x*A(x))).

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

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A214223 E.g.f. satisfies: A(x) = x/(1 - sinh(A(x))).

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