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A064310
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Generalized Catalan numbers C(-1; n).
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11
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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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See triangle A064334 with columns m built from C(-m; n), m >= 0, also for Derrida et al. references.
Unsigned sequence with a(0) := 0 is A000957 (Fine).
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LINKS
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FORMULA
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a(n) = Sum_{m=0..n-1} (-1)^m*(n-m)*binomial(n-1+m, m)/n.
a(n) = ((1/2)^n)*(1 + Sum_{k=0..n-1} C(k)*(-2)^k ), n >= 1, a(0)= 1, with C(n)=A000108(n) (Catalan).
G.f.: (1+x*c(-x)/2)/(1-x/2) = 1/(1-x*c(-x)) with c(x) g.f. of Catalan numbers A000108.
Conjecture: 2*n*a(n) + (7*n-12)*a(n-1) + 2*(-2*n+3)*a(n-2) = 0. - R. J. Mathar, Dec 02 2012
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MATHEMATICA
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a[n_]:= (1/2)^n*(1 + Sum[ CatalanNumber[k]*(-2)^k, {k, 0, n-1}]); Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 17 2013 *)
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PROG
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(PARI) {a(n) = (1 + sum(k=0, n-1, (-2)^k*binomial(2*k, k)/(k+1)))/2^n};
(Magma) [1] cat [(1 +(&+[(-2)^k*Binomial(2*k, k)/(k+1): k in [0..n-1]]))/2^n: n in [1..30]]; // G. C. Greubel, Feb 27 2019
(Sage) [1] + [(1 +sum((-2)^k*catalan_number(k) for k in (0..n-1)))/2^n for n in (1..30)] # G. C. Greubel, Feb 27 2019
(Python)
from itertools import count, islice
def A064310_gen(): # generator of terms
yield from (1, 1, 0)
a, c = 0, 1
for n in count(1):
yield (a:=(c:=c*((n<<2)+2)//(n+2))-a>>1)*(1 if n&1 else -1)
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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