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 A243626 #34 Nov 09 2024 07:02:04
%S A243626 0,1,4,22,142,1006,7570,59410,480910,3986230,33666154,288675322,
%T A243626 2506487158,21993277294,194717676514,1737297725602,15604975886110,
%U A243626 140999418473830,1280687414193370,11686769594020810,107093513405758342,985080052054562206,9092152494448328914
%N A243626 Expansion of (1-3*x-sqrt(x^2-10*x+1))/(4*x+2).
%H A243626 G. C. Greubel, <a href="/A243626/b243626.txt">Table of n, a(n) for n = 0..1000</a>
%F A243626 a(n) = (Sum_{k=0..n-1} (k+1)*Sum_{i=0..n-k-1} 2^i*binomial(n,n-k-i-1) * binomial(n+i-1,i))/n, n > 1, a(0)=0.
%F A243626 a(n) ~ sqrt(36+29*sqrt(6)) * (5+2*sqrt(6))^n / (50 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jun 08 2014
%F A243626 D-finite with recurrence: 2*n*a(n)-(30+19*n)*a(n+1)-(9+8*n)*a(n+2)+(n+3)*a(n+3)=0. - _Robert Israel_, Jan 07 2018
%F A243626 a(n) = Sum_{k=0..n-1} binomial(n-1,k)*hypergeom([k+1-n, n], [k+2], -2). - _Peter Luschny_, Jan 07 2018
%F A243626 O.g.f. A(x) = series reversion of x*(1 - x)/((1 + x)*(1 + 2*x)). - _Peter Bala_, Nov 08 2022
%p A243626 f:= gfun:-rectoproc({2*n*a(n)-(30+19*n)*a(n+1)-(9+8*n)*a(n+2)+(n+3)*a(n+3)=0,a(0)=0,a(1)=1,a(2)=4},a(n),remember):
%p A243626 map(f, [$0..30]); # _Robert Israel_, Jan 07 2018
%t A243626 CoefficientList[Series[(1-3*x-Sqrt[x^2-10*x+1])/(4*x+2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jun 08 2014 *)
%o A243626 (Maxima)
%o A243626 a(n):=sum((k+1)*sum(2^i*binomial(n,n-k-i-1)*binomial(n+i-1,i),i,0,n-k-1),k,0,n-1)/n;
%o A243626 (PARI) my(x='x+O('x^50)); concat([0], Vec((1-3*x - sqrt(x^2-10*x+1))/(4*x + 2))) \\ _G. C. Greubel_, Jun 02 2017
%Y A243626 Cf. A114710.
%K A243626 nonn,easy
%O A243626 0,3
%A A243626 _Vladimir Kruchinin_, Jun 08 2014