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A110556
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a(n) = binomial(2*n-1, n)*(-1)^n.
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8
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1, -1, 3, -10, 35, -126, 462, -1716, 6435, -24310, 92378, -352716, 1352078, -5200300, 20058300, -77558760, 300540195, -1166803110, 4537567650, -17672631900, 68923264410, -269128937220, 1052049481860, -4116715363800, 16123801841550
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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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E.g.f.: E(x) = 1 - x/(G(0)+2*x) ; G(k) = (k+1)^2 - 2*x*(2*k+1) + 2*x*(2*k+3)*((k+1)^2)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 21 2011
a(n) = coefficient of x^n of (1 / (1 + x))^n where 1 / (1 + x) is the g.f. of A033999. - Michael Somos, May 21 2013
E.g.f.: (1 + exp(-2*x) * BesselI(0,2*x)) / 2. - Ilya Gutkovskiy, Nov 03 2021
a(n) = binomial(-n, n).
O.g.f.: A(x) = (1 + sqrt(1 + 4*x))/(2*sqrt(1 + 4*x)) = 1/ (2 - c(-x)), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.
The g.f. A(x) satisfies A(x/(1 - x)^2) = 1/(1 + x). (End)
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EXAMPLE
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1 - x + 3*x^2 - 10*x^3 + 35*x^4 - 126*x^5 + 462*x^6 - 1716*x^7 + 6435*x^8 - ...
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MATHEMATICA
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Table[Binomial[2n-1, n](-1)^n, {n, 0, 30}] (* Harvey P. Dale, Apr 01 2012 *)
a[ n_] := (-1)^n Binomial[ 2 n - 1, n] (* Michael Somos, May 21 2013 *)
a[ n_] := SeriesCoefficient[(1 + x)^-n, {x, 0, n}] (* Michael Somos, May 21 2013 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( (1 + x)^-n + x * O(x^n), n))} /* Michael Somos, May 21 2013 */
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CROSSREFS
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Another version of A001700, which is the main entry.
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KEYWORD
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sign,easy,changed
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AUTHOR
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STATUS
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approved
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