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%I #19 Nov 04 2022 17:09:22
%S 1,1,12,145,1764,21602,266232,3301349,41178660,516512462,6513158376,
%T 82542517386,1051024082472,13442267711940,172638285341040,
%U 2225824753934445,28802104070304420,373966734921011990
%N Expansion of -(10*x + sqrt((1-10*x)*(1-14*x)))/(2*x).
%C Hankel transform is A077417.
%C The g.f. A(x) satisfies the continued fraction relation A(x) = 1/(1-x/(1-10*x-x*A(x))).
%F a(n) = sqrt(5/7) * 10^n * (6*hypergeom([1/2, n+1],[1],2/7)-7*hypergeom([1/2, n],[1],2/7)) / (n+1) for n > 0. - _Mark van Hoeij_, Jul 02 2010
%F D-finite with recurrence: (n+1)*a(n) +12*(1-2*n)*a(n-1) +140*(n-2)*a(n-2)=0. - _R. J. Mathar_, Sep 30 2012
%p with(LREtools): with(FormalPowerSeries): # requires Maple 2022
%p ogf:= -(10*x + sqrt((1-10*x)*(1-14*x)))/(2*x): req:= FindRE(ogf,x,u(n));
%p init:= [1, 1, 12, 145]: iseq:= seq(u(i-1)=init[i],i=1..nops(init)):
%p rmin:= subs(n=n-2, MinimalRecurrence(req,u(n),{iseq})[1]); # Mathar's recurrence
%p a:= gfun:-rectoproc({rmin, iseq}, u(n), remember):
%p seq(a(n),n=0..17); # _Georg Fischer_, Nov 03 2022
%p # Alternative, using function FindSeq from A174403:
%p ogf := -(10*x + sqrt((1-10*x)*(1-14*x)))/(2*x):
%p a := FindSeq(ogf): seq(a(n), n=0..17); # _Peter Luschny_, Nov 04 2022
%K nonn,easy
%O 0,3
%A _Paul Barry_, Mar 12 2010
%E Definiton corrected by _Peter Luschny_, Nov 05 2022
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