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%I #28 Apr 27 2023 11:22:58
%S 1,-1,0,-1,-2,-6,-18,-57,-186,-622,-2120,-7338,-25724,-91144,-325878,
%T -1174281,-4260282,-15548694,-57048048,-210295326,-778483932,
%U -2892818244,-10786724388,-40347919626,-151355847012,-569274150156
%N Expansion of 1/(1+x*c(x)), c(x) the g.f. of Catalan numbers A000108.
%C Hankel transform is (-1)^n.
%C Catalan transform of A033999. - _R. J. Mathar_, Nov 11 2008
%H Fung Lam, <a href="/A126983/b126983.txt">Table of n, a(n) for n = 0..1500</a>
%H Paul Barry, <a href="https://arxiv.org/abs/2107.00442">Conjectures and results on some generalized Rueppel sequences</a>, arXiv:2107.00442 [math.CO], 2021.
%F a(n) = (-1)^n*A064310(n).
%F a(n) = Sum_{k=0..n} A039599(n,k)*(-2)^k.
%F From _Philippe Deléham_, Nov 15 2009: (Start)
%F a(n) = Sum_{k=0..n} A106566(n,k)*(-1)^k, a(0)=1.
%F a(n) = -A000957(n) for n>0. (End)
%F Recurrence: 2*(n+2)*a(n+2) = (7*n+2)*a(n+1) + 2*(2*n+1)*a(n). - _Fung Lam_, May 07 2014
%F a(n) ~ -2^(2n)/sqrt(Pi*n^3)/9. - _Fung Lam_, May 07 2014
%t Table[(-1/2)^n*(1 + Sum[ CatalanNumber[k]*(-2)^k, {k, 0, n-1}]), {n, 0, 30}] (* _G. C. Greubel_, Feb 27 2019 *)
%o (PARI) {a(n) = (-1/2)^n*(1+sum(k=0,n-1, (-2)^k*binomial(2*k,k)/(k+1)))};
%o vector(30, n, n--; a(n)) \\ _G. C. Greubel_, Feb 27 2019
%o (Magma) [1] cat [(-1/2)^n*(1 +(&+[(-2)^k*Binomial(2*k,k)/(k+1): k in [0..n-1]])): n in [1..30]]; // _G. C. Greubel_, Feb 27 2019
%o (Sage) [1] + [(-1/2)^n*(1 +sum((-2)^k*catalan_number(k) for k in (0..n-1))) for n in (1..30)] # _G. C. Greubel_, Feb 27 2019
%o (Python)
%o from itertools import count, islice
%o def A126983_gen(): # generator of terms
%o yield from (1, -1, 0)
%o a, c = 0, 1
%o for n in count(1):
%o yield (a:=-a-(c:=c*((n<<2)+2)//(n+2))>>1)
%o A126983_list = list(islice(A126983_gen(),20)) # _Chai Wah Wu_, Apr 27 2023
%Y Cf. A000108, A000957, A039599, A064310, A106566.
%K sign
%O 0,5
%A _Philippe Deléham_, Mar 21 2007
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