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A126983
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Expansion of 1/(1+x*c(x)), c(x) the g.f. of Catalan numbers A000108.
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9
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1, -1, 0, -1, -2, -6, -18, -57, -186, -622, -2120, -7338, -25724, -91144, -325878, -1174281, -4260282, -15548694, -57048048, -210295326, -778483932, -2892818244, -10786724388, -40347919626, -151355847012, -569274150156
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OFFSET
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0,5
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COMMENTS
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Hankel transform is (-1)^n.
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} A039599(n,k)*(-2)^k.
a(n) = Sum_{k=0..n} A106566(n,k)*(-1)^k, a(0)=1.
Recurrence: 2*(n+2)*a(n+2) = (7*n+2)*a(n+1) + 2*(2*n+1)*a(n). - Fung Lam, May 07 2014
a(n) ~ -2^(2n)/sqrt(Pi*n^3)/9. - Fung Lam, May 07 2014
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MATHEMATICA
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Table[(-1/2)^n*(1 + Sum[ CatalanNumber[k]*(-2)^k, {k, 0, n-1}]), {n, 0, 30}] (* G. C. Greubel, Feb 27 2019 *)
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PROG
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(PARI) {a(n) = (-1/2)^n*(1+sum(k=0, n-1, (-2)^k*binomial(2*k, k)/(k+1)))};
(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
(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
(Python)
from itertools import count, islice
def A126983_gen(): # generator of terms
yield from (1, -1, 0)
a, c = 0, 1
for n in count(1):
yield (a:=-a-(c:=c*((n<<2)+2)//(n+2))>>1)
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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