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%I #19 Jun 03 2019 10:13:29
%S 1,1,-1,5,-25,141,-849,5349,-34825,232445,-1582081,10938709,-76616249,
%T 542472685,-3876400305,27919883205,-202480492905,1477306676445,
%U -10836099051105,79861379898165,-591082795606425
%N Generalized Catalan numbers C(-2; n).
%C See triangle A064334 with columns m built from C(-m; n), m >= 0, also for Derrida et al. references.
%F a(n) = sum((n-m)*binomial(n-1+m, m)*((-2)^m)/n, m=0..n-1) = ((1/3)^n)*(1+2*sum(C(k)*(-2*3)^k, k=0..n-1)), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
%F G.f.: (1+2*x*c(-2*x)/3)/(1-x/3) = 1/(1-x*c(-2*x)) with c(x) g.f. of Catalan numbers A000108.
%F a(n) = hypergeom([1-n, n], [-n], -2) for n>0. - _Peter Luschny_, Nov 30 2014
%F a(n) ~ -(-1)^n * 2^(3*n+1) / (25 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jun 03 2019
%t a[n_] := If[n==0, 1, Sum[(n-m)*Binomial[n+m-1, m]*(-2)^m/n, {m,0,n-1}]];
%t Table[a[n], {n,0,20}] (* _Jean-François Alcover_, Jun 03 2019 *)
%o (Sage)
%o import mpmath
%o mp.dps = 25; mp.pretty = True
%o a = lambda n: mpmath.hyp2f1(1-n, n, -n, -2) if n>0 else 1
%o [int(a(n)) for n in range(21)] # _Peter Luschny_, Nov 30 2014
%Y Cf. A064062.
%K sign,easy
%O 0,4
%A _Wolfdieter Lang_, Sep 21 2001
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