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A182626
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a(n) = Hypergeometric([-n, n], [1], 2).
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6
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1, -1, 5, -25, 129, -681, 3653, -19825, 108545, -598417, 3317445, -18474633, 103274625, -579168825, 3256957317, -18359266785, 103706427393, -586889743905, 3326741166725, -18885056428537, 107347191941249, -610916200215241
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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) = a(n). a(n) = (-1)^n * A002002(n) if n>0. a(n) = (-1)^n * A026003(2*n - 1) if n>0.
G.f.: 1 / ( 1 + x / (1 + 4*x / (1 - x^2 / (1 + 4*x / (1 - x^2 / (1 + 4*x / ...)))))). - Michael Somos, Jan 03 2013
a(n) = Sum_{k = 0..n} (-2)^k*binomial(n,k)*binomial(n+k-1,k) = (-1)^n*Sum_{k = 0..n-1} binomial(n,k+1)*binomial(n+k,k) = -Sum_{k = 0..n-1} (-2)^k*binomial(n-1,k)*binomial(n+k,k).
a(n) = -R(n-1,-2) for n >= 1, where R(n,x) denotes the n-th row polynomial of A178301.
a(n) = [x^n] ((x - 1)/(1 - 2*x))^n. Cf. A001003(n) = (-1)^(n+1)/(n+1)*[x^n] ((x - 1)/(1 - 2*x))^(n+1).
O.g.f.: 1/2 + 1/2*(1 + x)/sqrt(1 + 6*x + x^2).
exp( Sum_{n >= 1} a(n)*(-x)^n/n ) = 1 + x + 3*x^2 + 11*x^3 + 45*x^4 + ... is the o.g.f. for A001003.
Recurrence: n*(3 - 2*n )*a(n) = 2*(6*n^2 - 12*n + 5)*a(n-1) + (2*n - 1)*(n - 2)*a(n-2) with a(0) = 1, a(1) = -1. (End)
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EXAMPLE
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G.f. = 1 - x + 5*x^2 - 25*x^3 + 129*x^4 - 681*x^5 + 3653*x^6 - 19825*x^7 + ...
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MAPLE
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seq(simplify(hypergeom([-n, n], [1], 2)), n=0..21); # Peter Luschny, Mar 23 2015
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MATHEMATICA
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a[n_] := Hypergeometric2F1[ -n, n, 1, 2]; Array[a, 20, 0]
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PROG
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(PARI) {a(n) = sum( k=0, abs(n), 2^k * prod( i=0, k-1, i^2 - n^2 ) / k!^2)}
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/2 + 1/2*(1 + x)/Sqrt(1 + 6*x + x^2))); // G. C. Greubel, Aug 14 2018
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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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