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 A180254 #32 Jan 11 2018 04:57:51
%S A180254 1,1,5,47,655,12145,281645,7848575,255581095,9527142625,400145871125,
%T A180254 18700514207375,962618343061375,54121350164907025,3300097924150152125,
%U A180254 216926019220561025375,15292057853288541700375,1150861860384703732890625,92099021113080260667513125
%N A180254 E.g.f. A(x) satisfies: A'(x) = (1 - sqrt(1-4*A(x))) / (2*A(x)).
%H A180254 Vaclav Kotesovec, <a href="/A180254/b180254.txt">Table of n, a(n) for n = 1..350</a>
%F A180254 E.g.f. A(x) satisfies: A'(x) = 1/(1 - A(x)*A'(x)) = 1 + A(x)*A'(x)^2. - _Paul D. Hanna_, Jul 24 2012
%F A180254 E.g.f. A(x) satisfies: A'(x) = Sum_{n>=0} A000108(n)*A(x)^n where A000108(n) = binomial(2*n,n)/(n+1) forms the Catalan numbers. - _Paul D. Hanna_, Jul 24 2012
%F A180254 E.g.f.: Series_Reversion( Integral( dx/C(x) ) ) where C(x) = 1 + x*C(x)^2 is the Catalan function of A000108. - _Paul D. Hanna_, Dec 09 2013
%F A180254 Recurrence: 5*a(n) = 3*(18*n-35)*a(n-1) - 16*(3*n-8)*(3*n-7)*a(n-2). - _Vaclav Kotesovec_, Feb 19 2014
%F A180254 a(n) ~ 4 * (24/5)^(n-3/2) * n^(n-2) / exp(n). - _Vaclav Kotesovec_, Feb 19 2014
%e A180254 E.g.f.: A(x) = x + x^2/2! + 5*x^3/3! + 47*x^4/4! + 655*x^5/5! +...
%e A180254 where, by definition, A(x)*A'(x) = (1 - sqrt(1-4*A(x)))/2:
%e A180254 A(x)*A'(x) = x + 3*x^2/2! + 23*x^3/3! + 285*x^4/4! + 4885*x^5/5! +...
%t A180254 terms = 20; B[_] = 1; Do[B[x_] = (1 - Sqrt[1 - 4*Integrate[B[x], x]])/ (2*Integrate[B[x], x]) + O[x]^terms // Normal, terms];
%t A180254 A[x_] = Integrate[B[x], x];
%t A180254 CoefficientList[A[x]/x, x]*Range[terms]! (* _Jean-François Alcover_, Feb 21 2013, updated Jan 11 2018 *)
%t A180254 a[1] = a[2] = 1; a[n_] := a[n] = (1/5)*(-(144*n^2*a[n-2]) + 720*n*a[n-2] + 54*n*a[n-1] - 896*a[n-2] - 105*a[n-1]); Array[a, 20] (* _Jean-François Alcover_, Apr 21 2016 *)
%o A180254 (PARI) a(n)=local(A=x);for(i=1,n,A=intformal(1/(1-A*A'+x*O(x^n))));n!*polcoeff(A,n) \\ _Paul D. Hanna_, Jul 24 2012
%o A180254 for(n=1,20,print1(a(n),", "))
%o A180254 (PARI) a(n)=local(C=(1-sqrt(1-4*x+x^2*O(x^n)))/(2*x),A);n!*polcoeff(serreverse(intformal(1/C)),n)
%o A180254 for(n=1,20,print1(a(n),", ")) \\ _Paul D. Hanna_, Dec 09 2013
%Y A180254 Cf. A000108.
%K A180254 nonn
%O A180254 1,3
%A A180254 _Vladimir Kruchinin_, Jan 17 2011