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%I #27 Nov 20 2018 05:12:53
%S 1,1,1,-1,-3,1,9,1,-27,-13,81,67,-243,-285,729,1119,-2187,-4215,6561,
%T 15505,-19683,-56239,59049,202309,-177147,-724499,531441,2589521,
%U -1594323,-9254363,4782969,33111969,-14348907,-118725597,43046721
%N G.f.: A(x) = 1 + x/exp( Sum_{k>=1} (A((-1)^k*x) - 1)^k/k ).
%H Muniru A Asiru, <a href="/A157674/b157674.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.: A(x) = sqrt(1+4*x^2)/(sqrt(1+4*x^2) - x).
%F a(n) = Sum(k=1..n-1, (k*Sum(j=0..n, j*2^j*(-1)^j*binomial(n,j)* binomial(2*(n-k)-j-1,n-k-1)))/(n*(n-k)))+(-1)^(n-1) n>0, a(0)=1. - _Vladimir Kruchinin_, Apr 17 2011
%F a(2*n) = (-3)*a(2*n-2) = (-3)^(n-1), n >= 1; a(2*n+1) = (-3)*a(2*n-1) - 2*(-1)^n*A000108(n-1). - _Philippe Deléham_, Feb 02 2012
%F a(2*n+1) = (-1)^n * A137720(n). - _Vaclav Kotesovec_, Jul 31 2014
%e G.f.: A(x) = 1 + x + x^2 - x^3 - 3*x^4 + x^5 + 9*x^6 + x^7 - 27*x^8 - ...
%e ILLUSTRATION OF G.F.:
%e A(x) = 1 + x/exp((A(-x)-1) + (A(x)-1)^2/2 + (A(-x)-1)^3/3 + (A(x)-1)^4/4 + ...)
%e RELATED EXPANSION:
%e Coefficients of 1/A(x) include central binomial coefficients:
%e [1, -1, 0, 2, 0, -6, 0, 20, 0, -70, 0, 252, 0, -924, ...].
%e From _Philippe Deléham_, Feb 02 2012: (Start)
%e a(2) = 1,
%e a(4) = (-3)*1 = -3,
%e a(6) = (-3)*(-3) = 9,
%e a(8) = (-3)*9 = -27,
%e a(10) = (-3)*(-27) = 81,
%e a(12) = (3)*81 = -243, etc.
%e a(1) = 1,
%e a(3) = (-3)*1 + 2*1 = -1,
%e a(5) = (-3)*(-1)- 2*1 = 1,
%e a(7) = (-3)*1 + 2*2 = 1,
%e a(9) = (-3)*1 - 2*5 = -13,
%e a(11) = (-3)*(-13) + 2*14 = 67,
%e a(13) = (-3)*67 - 2*42 = -285,
%e a(15) = (-3)*(-285) + 2*132 = 1119, etc. (End)
%p 1,seq(sum(k*sum(j*2^j*(-1)^j*binomial(n,j)*binomial(2*(n-k)-j-1,n-k-1)/(n*(n-k)),j=0..n),k=1..n-1) +(-1)^(n-1),n=1..200); # _Muniru A Asiru_, Feb 04 2018
%t CoefficientList[Series[Sqrt[1+4*x^2]/(Sqrt[1+4*x^2] -x), {x, 0, 40}], x] (* _G. C. Greubel_, Nov 17 2018 *)
%o (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1,n,A=1+x*exp(-sum(k=1,n,(subst(A,x,(-1)^k*x+x*O(x^n))-1)^k/k))); polcoeff(A,n)}
%o (Maxima) a(n):=sum((k*sum(j*2^j*(-1)^j*binomial(n,j)*binomial(2*(n-k)-j-1,n-k-1),j,0,n))/(n*(n-k)),k,1,n-1)+(-1)^(n-1); /* _Vladimir Kruchinin_, Apr 17 2011 */
%o (GAP) Concatenation([1],List([1..200],n->Sum([1..n-1], k->k*Sum([0..n], j->j*2^j*(-1)^j*Binomial(n,j)*Binomial(2*(n-k)-j-1,n-k-1))/(n*(n-k)))+(-1)^(n-1))); # _Muniru A Asiru_, Feb 04 2018
%o (Sage) s= (sqrt(1+4*x^2)/(sqrt(1+4*x^2) - x)).series(x,40); s.coefficients(x, sparse=False) # _G. C. Greubel_, Nov 17 2018
%Y Cf. A000108, A000984, A156909, A137720.
%K sign
%O 0,5
%A _Paul D. Hanna_, Mar 05 2009
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