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 A136576 #20 Aug 03 2025 11:06:03
%S A136576 0,1,-3,10,-36,136,-532,2136,-8752,36448,-153824,656448,-2827904,
%T A136576 12281088,-53709632,236337536,-1045603072,4648306176,-20753783296,
%U A136576 93022530560,-418415228928,1888065744896,-8544699844608,38774062837760
%N A136576 Series reversion of x*c(x)/(1 - 2*x), c(x) the g.f. of A000108.
%C A136576 Hankel transform of a(n+1) is A136577 (conjecture).
%H A136576 G. C. Greubel, <a href="/A136576/b136576.txt">Table of n, a(n) for n = 0..1000</a>
%F A136576 G.f.: (sqrt(1+4*x-4*x^2)+4*x^2-2*x-1)/(8*x^2).
%F A136576 D-finite with recurrence (n+2)*a(n) + 2*(2*n+1)*a(n-1) + 4*(1-n)*a(n-2) = 0. - _R. J. Mathar_, Dec 11 2011
%F A136576 a(n) ~ (-1)^(n+1) * (3+2*sqrt(2)) * sqrt(4-2*sqrt(2)) * 2^(n-2) * (1+sqrt(2))^n / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jan 22 2014
%F A136576 For n >= 1, a(n) = (-1)^(n+1) * (1/2) * A071356(n) = (-1)^(n+1) * Sum_{k = 0..floor(n/2)} binomial(n, 2*k)*Catalan(k)*2^(n-k-1). The recurrence given above follows from this using the WZ algorithm. - _Peter Bala_, Apr 28 2024
%t A136576 CoefficientList[Series[(Sqrt[1+4*x-4*x^2]+4*x^2-2*x-1)/(8*x^2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jan 22 2014 *)
%o A136576 (PARI) x='x+O('x^50); concat([0], Vec((sqrt(1+4*x-4*x^2)+4*x^2-2*x-1)/(8*x^2))) \\ _G. C. Greubel_, Mar 21 2017
%Y A136576 Cf. A069731, A071356.
%K A136576 easy,sign
%O A136576 0,3
%A A136576 _Paul Barry_, Jan 08 2008