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%I #18 Jan 30 2020 21:29:15
%S 1,3,4,21,56,282,984,4813,19280,93150,403672,1945954,8845360,42766292,
%T 200419504,974134461,4659558048,22785183670,110564976792,543935554390,
%U 2667398588272,13196971915628,65238895435792,324431740601618,1614044041864800,8063536826420460
%N Expansion of (1 - sqrt( 1 - 4*x*sqrt( 1 + 4*x )) )/( 2*x ).
%H Alois P. Heinz, <a href="/A081698/b081698.txt">Table of n, a(n) for n = 0..500</a>
%F G.f.: (1-sqrt(1-4*x*sqrt(1+4*x)))/(2*x).
%F a(n) = sum(k=0..n, (binomial((k+1)/2,n-k)*binomial(2*k,k)*4^(n-k))/(k+1)). [_Vladimir Kruchinin_, Mar 13 2013]
%F D-finite with recurrence: n*(n+1)*a(n) +2*n*(5*n-7)*a(n-1) +4*(2*n^2-13*n+12)*a(n-2) -8*(2*n-3)*(14*n-37)*a(n-3) +16*(-64*n^2+392*n-573)*a(n-4) -96*(4*n-13)*(4*n-19)*a(n-5)=0. - _R. J. Mathar_, Jan 23 2020
%p a:= proc(n) option remember; `if`(n<4, [1, 3, 4, 21][n+1],
%p (2*n*(n+1)*(3-2*n) *a(n-1) +4*n*(2*n-1)*(2*n-3) *a(n-2)
%p +8*(2*n-3)*(8*n^2-16*n-15) *a(n-3)
%p +16*(4*n-15)*(4*n-9)*(n+1) *a(n-4)) /(n^2*(n+1)))
%p end:
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Mar 13 2013
%t a[n_] := Sum[Binomial[(k+1)/2, n-k]*Binomial[2*k, k]*4^(n-k)/(k+1), {k, 0, n}]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Apr 02 2015, after _Vladimir Kruchinin_ *)
%t CoefficientList[Series[(1-Sqrt[1-4x Sqrt[1+4x]])/(2x),{x,0,30}],x] (* _Harvey P. Dale_, Oct 30 2017 *)
%Y Cf. A081696.
%K easy,nonn
%O 0,2
%A _Emanuele Munarini_, Apr 02 2003
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