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 A214770 #11 Jan 13 2014 09:13:46
%S A214770 1,2,12,120,1680,30232,664832,17277120,518031360,17602865312,
%T A214770 668505311232,28059791760000,1289932186583040,64455076284318592,
%U A214770 3478305412257677312,201608948937441269760,12491465252403224248320,823886511479340063068672,57633367371058675735068672
%N A214770 E.g.f. satisfies: A(x) = x + sin(A(x))*sinh(A(x)).
%F A214770 E.g.f. satisfies:
%F A214770 (1) A(x) = Series_Reversion(x - sin(x)*sinh(x)).
%F A214770 (2) A(x) = x + Sum_{n>=1} (-1)^(n-1)*2^(2*n-1) * A(x)^(4*n-2)/(4*n-2)!.
%F A214770 (3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) sin(x)^n*sinh(x)^n / n!.
%F A214770 (4) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) sin(x)^n*sinh(x)^n/x / n! ).
%F A214770 a(n) ~ n^(n-1) / (exp(n) * r^(n-1/2) * sqrt(2*cos(s)*cosh(s))), where s = 0.50105258964301589... is the root of the equation cosh(s)*sin(s) + cos(s)*sinh(s) = 1, and r = s - sin(s)*sinh(s) = 0.25017469884019539.... - _Vaclav Kotesovec_, Jan 13 2014
%e A214770 E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1680*x^5/5! + 30232*x^6/6! +...
%e A214770 where A(x - sin(x)*sinh(x)) = x and A(x) = x + sin(A(x))*sinh(A(x)).
%e A214770 Related expansions:
%e A214770 sin(x)*sinh(x) = 2*x^2/2! - 8*x^6/6! + 32*x^10/10! - 128*x^14/14! + 512*x^18/18! -+...+ (-1)^(n-1)*2^(2*n-1)*x^(4*n-2)/(4*n-2)! +-...
%e A214770 sin(A(x)) = x + 2*x^2/2! + 11*x^3/3! + 108*x^4/4! + 1501*x^5/5! + 26902*x^6/6! +...
%e A214770 sinh(A(x)) = x + 2*x^2/2! + 13*x^3/3! + 132*x^4/4! + 1861*x^5/5! + 33622*x^6/6! +...
%e A214770 Other series:
%e A214770 A(x) = x + sin(x)*sinh(x) + d/dx sin(x)^2*sinh(x)^2/2! + d^2/dx^2 sin(x)^3*sinh(x)^3/3! + d^3/dx^3 sin(x)^4*sinh(x)^4/4! +...
%e A214770 log(A(x)/x) = sin(x)*sinh(x)/x + d/dx sin(x)^2*sinh(x)^2/x/2! + d^2/dx^2 sin(x)^3*sinh(x)^3/x/3! + d^3/dx^3 sin(x)^4*sinh(x)^4/x/4! +...
%t A214770 Rest[CoefficientList[InverseSeries[Series[x - Sin[x]*Sinh[x], {x, 0, 20}], x],x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Jan 13 2014 *)
%o A214770 (PARI) {a(n)=n!*polcoeff(serreverse(x-sin(x+x*O(x^n))*sinh(x+x*O(x^n))), n)}
%o A214770 for(n=1, 25, print1(a(n), ", "))
%o A214770 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A214770 {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, sin(x+x*O(x^n))^m*sinh(x+x*O(x^n))^m/m!)); n!*polcoeff(A, n)}
%o A214770 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A214770 {a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, sin(x+x*O(x^n))^m*sinh(x+x*O(x^n))^m/x/m!))); n!*polcoeff(A, n)}
%Y A214770 Cf. A143134.
%K A214770 nonn
%O A214770 1,2
%A A214770 _Paul D. Hanna_, Jul 31 2012