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

Author

Paul D. Hanna, Jul 07 2012

Keywords

• nonn

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

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

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