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A179533 Expansion of (1/(1-x-2x^2))*c(x/(1-x-2x^2)), c(x) the g.f. of A000108.

%I A179533 #21 Nov 04 2022 10:46:53
%S A179533 1,2,7,23,85,332,1369,5870,25945,117374,540805,2528675,11966923,
%T A179533 57206972,275824159,1339721519,6549093013,32195473406,159065828029,
%U A179533 789395034701,3933239089903,19668745466636,98679891233803,496570499905832,2505670304785615,12675395921692394,64270076976110203,326580624341708693,1662796531746045157,8481930651824392268,43341418581113085697
%N A179533 Expansion of (1/(1-x-2x^2))*c(x/(1-x-2x^2)), c(x) the g.f. of A000108.
%C A179533 Hankel transform is A168495(n+1).
%F A179533 G.f.: 1/(1-x-2x^2-x/(1-x/(1-x-2x^2-x/(1-x/(1-x-2x^2-x/(1-x/(1-x-2x^2-x/(1-x/(1-... (continued fraction);
%F A179533 a(n) = Sum_{k=0..n} A000108(k)*Sum_{j=0..n-k} C(k+j,k)*C(j,n-k-j)*2^(n-k-j).
%F A179533 a(n) = Sum_{k=0..n} A073370(n,k)*A000108(k).
%F A179533 D-finite with recurrence: (n+1)*a(n) +2*(1-3n)*a(n-1) +(n-1)*a(n-2) +4*(3n-5)*a(n-3) +4*(n-3)*a(n-4)= 0. - _R. J. Mathar_, Nov 17 2011
%p A179533 with(LREtools): with(FormalPowerSeries): # requires Maple 2022
%p A179533 ogf:= (1/(1-x-2*x^2))*(1 - sqrt(1 - 4*(x/(1-x-2*x^2)))) / (2*(x/(1-x-2*x^2))):
%p A179533 init:= [1, 2, 7, 23, 85, 332, 1369];
%p A179533 iseq:= seq(u(i-1)=init[i],i=1..nops(init)): req:= FindRE(ogf,x,u(n));
%p A179533 rmin:= subs(n=n-4,MinimalRecurrence(req,u(n),{iseq})[1]); # Mathar's recurrence
%p A179533 a:= gfun:-rectoproc({rmin, iseq}, u(n), remember):
%p A179533 seq(a(n),n=0..30); # _Georg Fischer_, Nov 04 2022
%Y A179533 Cf. A000108, A073370, A168495.
%K A179533 nonn
%O A179533 0,2
%A A179533 _Paul Barry_, Jan 08 2011