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A064306
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Convolution of A052701 (Catalan numbers multiplied by powers of 2) with powers of -1.
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5
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1, 1, 7, 33, 191, 1153, 7295, 47617, 318463, 2170881, 15028223, 105365505, 746651647, 5339185153, 38478839807, 279201841153, 2037998419967, 14954803494913, 110255315877887, 816299567480833, 6066679566041087
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = (-1)^n*Sum_{k=0,..,n} (C(k)/(-1/2)^k) with C(k)=A000108(k) (Catalan).
a(n) = -a(n-1) + C(n)*2^n, n >= 0, a(-1) := 0, with C(n)=A000108(n).
G.f.: A(2*x)/(1+x), with A(x) g.f. of Catalan numbers A000108.
Recurrence: (n+1)*a(n) = (7*n-5)*a(n-1) + 4*(2*n-1)*a(n-2). - Vaclav Kotesovec, Dec 09 2013
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MATHEMATICA
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CoefficientList[Series[(1-Sqrt[1-8*x])/(4*x*(1+x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 09 2013 *)
Table[FullSimplify[2^(n+1)*(2*n+2)! * Hypergeometric2F1Regularized[1, n+3/2, n+3, -8]/(n+1)! + (-1)^n/2], {n, 0, 20}] (* Vaclav Kotesovec, Dec 09 2013 *)
Table[(-1)^n*Sum[(-2)^k * CatalanNumber[k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Jan 27 2017 *)
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PROG
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(Sage)
f, c, n = 1, 1, 1
while True:
yield f
n += 1
c = c * (8*n - 12) // n
f = c - f
(PARI) for(n=0, 25, print1((-1)^n*sum(k=0, n, (-2)^k*binomial(2*k, k)/(k+1)), ", ")) \\ G. C. Greubel, Jan 27 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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