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%I #14 Sep 01 2024 17:12:19
%S 1,3,15,78,411,2178,11574,61596,328083,1748346,9319650,49688004,
%T 264943902,1412826132,7534329804,40180417848,214285867011,
%U 1142822599722,6094924220394,32505784870068,173362418710506,924593002335612,4931138212856244,26299312319336328,140262655976846766
%N Expansion of (1+sqrt(1-4x))/(4sqrt(1-4x)-2).
%C Apply the Riordan matrix ((1+sqrt(1-4x))/2,(1-sqrt(1-4x))/2) (inverse of (1/(1-x),x(1-x))) to 4^n.
%H Vincenzo Librandi, <a href="/A104530/b104530.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = 0^n + Sum_{k=0..n} 3^(k+1)*C(2*n-1, n-k-1)*2*(k+1)/(n+k+1).
%F Recurrence: 3*n*a(n) = 2*(14*n-9)*a(n-1) - 32*(2*n-3)*a(n-2). - _Vaclav Kotesovec_, Oct 17 2012
%F a(n) ~ 2^(4*n-1)/3^n. - _Vaclav Kotesovec_, Oct 17 2012
%t CoefficientList[Series[(1+Sqrt[1-4*x])/(4*Sqrt[1-4*x]-2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 17 2012 *)
%o (PARI) x='x+O('x^66); Vec((1+sqrt(1-4*x))/(4*sqrt(1-4*x)-2)) \\ _Joerg Arndt_, May 13 2013
%Y Cf. A088218, A067336, A076025.
%K easy,nonn,changed
%O 0,2
%A _Paul Barry_, Mar 12 2005
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