|
%I #19 Jan 30 2020 21:29:17
%S 1,0,1,2,3,8,17,38,91,212,509,1234,3007,7408,18353,45742,114643,
%T 288620,729749,1852138,4716951,12050920,30876185,79317990,204256027,
%U 527171556,1363428637,3533070818,9171798815,23849951200,62116162081
%N Expansion of (1-sqrt(1-4*(x+x^2)^2))/(2*(x+x^2)^2).
%H G. C. Greubel, <a href="/A256169/b256169.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{k=floor((n-1)/4)..(n-1)} binomial(2*k,n-2*k-1)*C(k)}, where C(k) are the Catalan numbers (A000108).
%F G.f. g(x) satisfies 1 - g(x) + x^2 (1+x)^2 g(x)^2 = 0. Recurrence: a(n) = sum(j>=0, a(j)*(a(n-j-2)+2*a(n-j-3)+a(n-j-4))) for n >= 1, where a(j) = 0 for j < 0. - _Robert Israel_, Mar 17 2015
%F D-finite with recurrence: (n+2)*a(n) = -(n+3)*a(n-1) + 4*(n-1)*a(n-2) + 4*(3*n - 5)*a(n-3) + 4*(3*n - 7)*a(n-4) + 4*(n-3)*a(n-5). - _Vaclav Kotesovec_, Mar 17 2015
%F a(n) ~ sqrt(6-2*sqrt(3)) * (1+sqrt(3))^n / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Mar 17 2015
%p f:= proc(n) option remember;
%p add(procname(j)*(procname(n-j-2)+2*procname(n-j-3)+procname(n-j-4)),j=0..n-2)
%p end proc:
%p f(0):= 1: f(-1):= 0: f(-2):= 0:
%p seq(f(n),n=0..100); # _Robert Israel_, Mar 17 2015
%t CoefficientList[Series[(1-Sqrt[1-4*(x+x^2)^2])/(2*(x+x^2)^2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 17 2015 *)
%o (Maxima)
%o a(n):=sum((binomial(2*k,n-2*k-1)*binomial(2*k,k))/(k+1),k,floor((n-1)/4),n-1);
%o (PARI) default(seriesprecision, 50); Vec((1-sqrt(1-4*(x+x^2)^2))/(2*(x+x^2)^2) + O(x^50)); \\ _Michel Marcus_, Mar 17 2015
%Y Cf. A000108.
%K nonn
%O 0,4
%A _Vladimir Kruchinin_, Mar 17 2015
|
