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%I #35 Sep 08 2022 08:45:55
%S 1,-1,5,-25,129,-681,3653,-19825,108545,-598417,3317445,-18474633,
%T 103274625,-579168825,3256957317,-18359266785,103706427393,
%U -586889743905,3326741166725,-18885056428537,107347191941249,-610916200215241
%N a(n) = Hypergeometric([-n, n], [1], 2).
%H Vincenzo Librandi, <a href="/A182626/b182626.txt">Table of n, a(n) for n = 0..1000</a>
%F a(-n) = a(n). a(n) = (-1)^n * A002002(n) if n>0. a(n) = (-1)^n * A026003(2*n - 1) if n>0.
%F 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
%F a(n) = (-1)^n*Sum_{k=0..n} A253283(n,k). - _Peter Luschny_, Mar 23 2015
%F From _Peter Bala_, Jun 17 2015: (Start)
%F 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).
%F a(n) = -R(n-1,-2) for n >= 1, where R(n,x) denotes the n-th row polynomial of A178301.
%F 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).
%F O.g.f.: 1/2 + 1/2*(1 + x)/sqrt(1 + 6*x + x^2).
%F 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.
%F 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)
%e G.f. = 1 - x + 5*x^2 - 25*x^3 + 129*x^4 - 681*x^5 + 3653*x^6 - 19825*x^7 + ...
%p seq(simplify(hypergeom([-n, n],[1],2)), n=0..21); # _Peter Luschny_, Mar 23 2015
%t a[n_] := Hypergeometric2F1[ -n, n, 1, 2]; Array[a, 20, 0]
%o (PARI) {a(n) = sum( k=0, abs(n), 2^k * prod( i=0, k-1, i^2 - n^2 ) / k!^2)}
%o (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
%Y Cf. A002002, A026003, A253283, A001003, A178301.
%K sign,easy
%O 0,3
%A _Michael Somos_, Feb 06 2011
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