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 A095839 #25 Aug 05 2025 22:56:23
%S A095839 1,1,5,51,807,17445,479565,16019955,630301455,28552506885,
%T A095839 1463744449125,83780913568275,5296205435649975,366478026602012325,
%U A095839 27552067849812030525,2236327624673777509875,194908916445067162713375,18154937081288124469477125
%N A095839 a(n) = ((2*n)!/(n!*2^(n-1)))*Integral_{x=1/2..1} (sqrt(1-x^2)/x)^(2*n) dx.
%C A095839 From _Paul Hanna_, Dec 22 2013: (Start)
%C A095839 E.g.f. satisfies: A(x) = 1 + A(x)^3 * Integral 1/A(x) dx. Compare to: G(x) = 1 + G(x)^3 * Integral 1/G(x)^3 dx, where G(x)-1 is the e.g.f. of A058562, the number of 3-way series-parallel networks with n labeled edges.
%C A095839 A(x)^3 = 1 + 3*x + 21*x^2/2! + 249*x^3/3! + 4275*x^4/4! + 97155*x^5/5! +...
%C A095839 Integral 1/A(x) dx = x - x^2/2! - 3*x^3/3! - 27*x^4/4! - 405*x^5/5! - 8505*x^6/6! +...
%C A095839 Series_Reversion(Integral 1/A(x) dx) = x + x^2/2 + 3*x^3/3 + 12*x^4/4 + 55*x^5/5 + 273*x^6/6 + 1428*x^7/7 +...+ A001764(n-1)*x^n/n +...
%C A095839 E.g.f.: 1 + Series_Reversion( (x - x^2/2) / (1+x)^2 ). (End)
%H A095839 Robert Israel, <a href="/A095839/b095839.txt">Table of n, a(n) for n = 0..347</a>
%F A095839 D-finite with a(n) +(-4*n+9)*a(n-1) -6*(n-1)*(2*n-3)*a(n-2)=0. - _R. J. Mathar_, Feb 13 2014
%F A095839 E.g.f.: (2-sqrt(1-6*x))/(1+2*x). Recurrence follows from the d.e. (12*x^2+4*x-1)*y''+(30*x-1)*y'+6*y=0 satisfied by this. - _Robert Israel_, May 08 2018
%F A095839 a(n) ~ 2^(n-5/2) * 3^(n+1) * n^(n-1) / exp(n). - _Vaclav Kotesovec_, Dec 27 2013
%p A095839 A095839 := proc(n)
%p A095839 local k;
%p A095839 (4^k-2)/2/(2*k-1) ;
%p A095839 add(%*(-1)^k*binomial(n,k),k=0..n) ;
%p A095839 %*(-1)^n*(2*n)!/n!/2^(n-1) ;
%p A095839 end proc: # _R. J. Mathar_, Feb 13 2014
%t A095839 f[n_] := Numerator[ Integrate[(Sqrt[1 - x^2]/x)^(2n), {x, 1/2, 1}]*(2n)!/(n!2^(n + 1)!)]; Table[ f[n], {n, 0, 11}] (* _Robert G. Wilson v_ *)
%t A095839 f[n_] := Numerator[2^(-2 - Gamma[2 + n])*3^(1 + n)*(2*n)!* Hypergeometric2F1Regularized[1, 1/2 + n, 2 + n, -3]]; Table[f[n], {n, 0, 11}] (* _Eric W. Weisstein_, Nov 19 2005 *)
%o A095839 (PARI)
%o A095839 {a(n)=local(A=(2-sqrt(1-6*x+x^2*O(x^n)))/(1+2*x)); n!*polcoeff(A, n)}
%o A095839 for(n=0, 25, print1(a(n), ", ")) \\ _Paul D. Hanna_, Dec 22 2013
%Y A095839 Cf. A058562, A234289, A234291, A001764, A000108.
%K A095839 nonn
%O A095839 0,3
%A A095839 Al Hakanson (Hawkuu(AT)excite.com), Jun 08 2004
%E A095839 a(8)-a(11) from _Robert G. Wilson v_, Nov 18 2005
%E A095839 Definition corrected by _Robert Israel_, May 08 2018