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%I #15 Jan 30 2020 21:29:16
%S 1,2,8,36,180,956,5300,30316,177604,1060284,6427092,39452364,
%T 244748196,1532044572,9664688436,61380865452,392148430212,
%U 2518518772604,16250624534420,105297028489612,684865176181348
%N Expansion of (1-x-sqrt(1-6*x-7*x^2))/(2*(1+2*x)).
%C Series reversion of x(1-x)/(1+x+2x^2).
%C Hankel transform is 4^C(n+1,2)=A053763(n+1).
%H Vincenzo Librandi, <a href="/A129148/b129148.txt">Table of n, a(n) for n = 1..200</a>
%F a(n)=sum{k=0..n, sum{j=0..n, C(n,j)*C(n-k,j+k-n)*C(n-k)*3^(j+k-n)}}, C(n)=A000108(n); a(n)=(1/(2*pi))*int(x^n*sqrt(7+6*x-x^2)/(2+x),x,-1,7);
%F D-finite with recurrence: n*a(n) = (4*n-9)*a(n-1) + (19*n-39)*a(n-2) + 14*(n-3)*a(n-3) . - _Vaclav Kotesovec_, Oct 20 2012
%F a(n) ~ 7^(n+1/2)/(9*sqrt(2*Pi)*n^(3/2)) . - _Vaclav Kotesovec_, Oct 20 2012
%t Rest[CoefficientList[Series[(1-x-Sqrt[1-6*x-7*x^2])/(2*(1+2*x)), {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Oct 20 2012 *)
%t Table[(1/(2*Pi))*Integrate[x^n*Sqrt[7+6*x-x^2]/(2+x),{x,-1,7}],{n,0,10}] (* _Vaclav Kotesovec_, Oct 20 2012 *)
%K easy,nonn
%O 1,2
%A _Paul Barry_, Apr 01 2007
%E Offset corrected to 1, _Vaclav Kotesovec_, Oct 20 2012
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