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A104496 Expansion of 2*(2*x+1)/((x+1)*(sqrt(4*x+1)+1)).

%I A104496 #20 Jan 30 2020 21:29:15
%S A104496 1,0,0,-1,5,-19,67,-232,804,-2806,9878,-35072,125512,-452388,1641028,
%T A104496 -5986993,21954973,-80884423,299233543,-1111219333,4140813373,
%U A104496 -15478839553,58028869153,-218123355523,821908275547,-3104046382351,11747506651599,-44546351423299,169227201341651
%N A104496 Expansion of 2*(2*x+1)/((x+1)*(sqrt(4*x+1)+1)).
%C A104496 Previous name was: Row sums of triangle A104495. A104495 equals the matrix inverse of triangle A099602, where row n of A099602 equals the inverse Binomial transform of column n of the triangle of trinomial coefficients (A027907).
%C A104496 Absolute row sums of triangle A104495 forms A014137 (partial sums of Catalan numbers).
%H A104496 G. C. Greubel, <a href="/A104496/b104496.txt">Table of n, a(n) for n = 0..1000</a>
%F A104496 G.f.: A(x) = (1 + 2*x)/(1+x)/(1+x - x^2*Catalan(-x)^2), where Catalan(x)=(1-(1-4*x)^(1/2))/(2*x) (cf. A000108).
%F A104496 a(n) ~ (-1)^n * 2^(2*n+1) / (3*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 06 2014
%F A104496 D-finite with recurrence: (n+1)*a(n) +(7*n-3)*a(n-1) +2*(7*n-12)*a(n-2) +4*(2*n-5)*a(n-3)=0. - _R. J. Mathar_, Jan 23 2020
%p A104496 gf := (2*(2*x+1))/((x+1)*(sqrt(4*x+1)+1)): ser := series(gf,x,30):
%p A104496 seq(coeff(ser,x,n),n=0..28); # _Peter Luschny_, Apr 25 2016
%t A104496 CoefficientList[Series[(1+2*x)/(1+x)/(1+x - (1-(1+4*x)^(1/2))^2/4), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 06 2014 *)
%o A104496 (PARI) {a(n)=local(X=x+x*O(x^n));polcoeff( (1+2*X)/(1+X)/(1+X-(1-(1+4*X)^(1/2))^2/4),n,x)}
%o A104496 (Python)
%o A104496 from itertools import accumulate
%o A104496 def A104496_list(size):
%o A104496     if size < 1: return []
%o A104496     L, accu = [1], [1]
%o A104496     for n in range(size-1):
%o A104496         accu = list(accumulate(accu + [-accu[0]]))
%o A104496         L.append(-(-1)**n*accu[-1])
%o A104496     return L
%o A104496 print(A104496_list(29)) # _Peter Luschny_, Apr 25 2016
%Y A104496 Cf. A104495, A099602, A027907, A000108.
%K A104496 sign,easy
%O A104496 0,5
%A A104496 _Paul D. Hanna_, Mar 11 2005
%E A104496 New name using the g.f. of the author by _Peter Luschny_, Apr 25 2016