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.
%I A214225 #24 Nov 16 2017 02:36:49
%S A214225 1,2,12,112,1440,23616,471296,11085824,300349440,9211187200,
%T A214225 315448860672,11932326789120,494098626904064,22230301612703744,
%U A214225 1079857012109475840,56326462301645307904,3140024293968001892352,186308007164786201591808,11722541029509094870876160
%N A214225 E.g.f. satisfies: A(x) = x/(1 - tanh(A(x))).
%H A214225 G. C. Greubel, <a href="/A214225/b214225.txt">Table of n, a(n) for n = 1..370</a>
%F A214225 E.g.f. A(x) satisfies:
%F A214225 (1) A(x - x*tanh(x)) = x.
%F A214225 (2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*tanh(x)^n/n!.
%F A214225 (3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*tanh(x)^n/n! ).
%F A214225 (4) A(x) = Sum_{n>=1} n^(n-1) * cosh(n*x) * x^n / n!. - _Paul D. Hanna_, Nov 20 2012
%F A214225 (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
%F A214225 a(n) = n*A201595(n-1).
%F A214225 a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*k^(n-1).
%F A214225 a(n) = (n-1)! * [x^n] x/(1 - tanh(x))^n.
%F A214225 a(n) = A038049(n)/2. - _R. J. Mathar_, _Peter Bala_, Mar 24 2013
%F A214225 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
%e A214225 E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! +...
%e A214225 Related expansions:
%e A214225 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! +...
%e A214225 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! +...
%e A214225 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! +...
%e A214225 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! +...
%t A214225 Rest[CoefficientList[InverseSeries[Series[x-x*Tanh[x],{x,0,20}],x],x]*Range[0,20]!] (* _Vaclav Kotesovec_, Sep 17 2013 *)
%t A214225 Flatten[{1,Table[1/2*Sum[Binomial[n,k]*k^(n-1),{k,0,n}],{n,2,20}]}] (* _Vaclav Kotesovec_, Sep 17 2013 *)
%o A214225 (PARI) {a(n)=(1/2)*sum(k=0,n,binomial(n,k)*k^(n-1))}
%o A214225 for(n=1, 25, print1(a(n), ", "))
%o A214225 (PARI) {a(n)=(n-1)!*polcoeff(x/(1 - tanh(x+x*O(x^n)))^n,n)}
%o A214225 (PARI) {a(n)=n!*polcoeff(serreverse(x-x*tanh(x+x*O(x^n))), n)}
%o A214225 (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
%o A214225 for(n=1, 25, print1(a(n), ", "))
%o A214225 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A214225 {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)}
%o A214225 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A214225 {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)}
%Y A214225 Cf. A201595, A202617, A214222, A214223, A214224.
%K A214225 nonn
%O A214225 1,2
%A A214225 _Paul D. Hanna_, Jul 07 2012