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A064311 Generalized Catalan numbers C(-2; n).

%I A064311 #23 Sep 13 2024 08:08:41
%S A064311 1,1,-1,5,-25,141,-849,5349,-34825,232445,-1582081,10938709,-76616249,
%T A064311 542472685,-3876400305,27919883205,-202480492905,1477306676445,
%U A064311 -10836099051105,79861379898165,-591082795606425
%N A064311 Generalized Catalan numbers C(-2; n).
%C A064311 See triangle A064334 with columns m built from C(-m; n), m >= 0, also for Derrida et al. references.
%F A064311 a(n) = (1/n) * Sum_{m = 0..n-1} (n-m)*binomial(n-1+m, m)*(-2)^m = ((1/3)^n)*(1 + 2*Sum_{k = 0..n-1} C(k)*(-2*3)^k), for n >= 1, with a(0) := 1, and where C(n) = A000108(n), the Catalan numbers.
%F A064311 G.f.: (1+2*x*c(-2*x)/3)/(1-x/3) = 1/(1-x*c(-2*x)) with c(x) the g.f. of the Catalan numbers A000108.
%F A064311 a(n) = hypergeom([1-n, n], [-n], -2) for n>0. - _Peter Luschny_, Nov 30 2014
%F A064311 a(n) ~ -(-1)^n * 2^(3*n+1) / (25 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jun 03 2019
%F A064311 G.f. A(x) = 1 + series_reversion(x*(1 - (m-1)*x)/(1 + x)^2) at m = -2. - _Peter Bala_, Sep 08 2024
%t A064311 a[n_] := If[n==0, 1, Sum[(n-m)*Binomial[n+m-1, m]*(-2)^m/n, {m,0,n-1}]];
%t A064311 Table[a[n], {n,0,20}] (* _Jean-François Alcover_, Jun 03 2019 *)
%o A064311 (Sage)
%o A064311 import mpmath
%o A064311 mp.dps = 25; mp.pretty = True
%o A064311 a = lambda n: mpmath.hyp2f1(1-n, n, -n, -2) if n>0 else 1
%o A064311 [int(a(n)) for n in range(21)] # _Peter Luschny_, Nov 30 2014
%Y A064311 Generalized Catalan numbers C(m; n): A000012 (m = 0), A000108 (m = 1), A064062 (m = 2), A064063 (m = 3), A064087 - A064093 (m = 4 thru 10); A064310 (m = -1) and A064325 - A064333 (m = -3 thru -11).
%K A064311 sign,easy
%O A064311 0,4
%A A064311 _Wolfdieter Lang_, Sep 21 2001