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A108624 G.f. satisfies x = A(x)*(1+A(x))/(1-A(x)-(A(x))^2).

%I A108624 #20 Oct 20 2023 01:46:52
%S A108624 1,0,-1,1,1,-4,3,8,-23,10,67,-153,9,586,-1081,-439,5249,-7734,-7941,
%T A108624 47501,-53791,-105314,430119,-343044,-1249799,3866556,-1730017,
%U A108624 -13996097,34243897,-1947204,-150962373,296101864,121857185
%N A108624 G.f. satisfies x = A(x)*(1+A(x))/(1-A(x)-(A(x))^2).
%C A108624 Row sums of triangle A202327. - _Peter Luschny_, Apr 26 2017
%H A108624 G. C. Greubel, <a href="/A108624/b108624.txt">Table of n, a(n) for n = 1..1000</a>
%F A108624 a(n) = (1/n)*Sum_{k=1..n} ( k * Sum_{j=0..n} (-1)^(k+j)*binomial(j, 2*j-n-k)*binomial(n,j) ). - _Vladimir Kruchinin_, May 19 2012
%F A108624 G.f.: (-1 + x + sqrt(1+2*x+5*x^2))/(2*(1+x)). - _G. C. Greubel_, Oct 20 2023
%t A108624 a[n_]:= Sum[k Sum[(-1)^(j-k) Binomial[j, 2j-n-k] Binomial[n, j], {j, 0, n}], {k, 1, n}]/n;
%t A108624 Array[a, 33] (* _Jean-François Alcover_, Jun 13 2019, after _Vladimir Kruchinin_ *)
%t A108624 Rest@CoefficientList[Series[(-1+x+Sqrt[1+2*x+5*x^2])/(2*(1+x)), {x,0, 41}], x] (* _G. C. Greubel_, Oct 20 2023 *)
%o A108624 (Julia)
%o A108624 function A108624_list(len::Int)
%o A108624     len <= 0 && return BigInt[]
%o A108624     T = zeros(BigInt, len, len); T[1,1] = 1
%o A108624     S = Array(BigInt, len); S[1] = 1
%o A108624     for n in 2:len
%o A108624         T[n,n] = 1
%o A108624         for k in 1:n-1
%o A108624             T[n,k] = (k > 1 ? T[n-1,k-1] : 0) - T[n-1,k] - T[n-1,k+1]
%o A108624         end
%o A108624         S[n] = sum(T[n,k] for k in 1:n)
%o A108624     end
%o A108624 S end
%o A108624 println(A108624_list(33)) # _Peter Luschny_, Apr 27 2017
%o A108624 (Magma)
%o A108624 R<x>:=PowerSeriesRing(Rationals(), 41);
%o A108624 Coefficients(R!( (-1+x+Sqrt(1+2*x+5*x^2))/(2*(1+x)) )); // _G. C. Greubel_, Oct 20 2023
%o A108624 (SageMath)
%o A108624 def A108624_list(prec):
%o A108624     P.<x> = PowerSeriesRing(ZZ, prec)
%o A108624     return P( (-1+x+sqrt(1+2*x+5*x^2))/(2*(1+x))).list()
%o A108624 a=A108624_list(41); a[1:] # _G. C. Greubel_, Oct 20 2023
%Y A108624 Cf. A039980, A202327.
%Y A108624 Except for signs, same as A108623.
%K A108624 sign
%O A108624 1,6
%A A108624 _Christian G. Bower_, Jun 12 2005