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A064325 Generalized Catalan numbers C(-3; n). 6

%I

%S 1,1,-2,13,-98,826,-7448,70309,-686090,6865150,-70057772,726325810,

%T -7628741204,81002393668,-868066319108,9376806129493,-101988620430938,

%U 1116026661667318,-12277755319108748,135715825209716038,-1506587474535945788,16789107646422189868,-187747069029477151328

%N Generalized Catalan numbers C(-3; 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)*((-3)^m)/n, m=0..n-1) = ((1/4)^n)*(1+3*sum(C(k)*(-3*4)^k, k=0..n-1)), n >= 1, a(0) = 1; with C(n) = A000108(n) (Catalan).

%F G.f.: (1+3*x*c(-3*x)/4)/(1-x/4) = 1/(1-x*c(-3*x)) with c(x) g.f. of Catalan numbers A000108.

%F a(n) = hypergeometric([1-n, n], [-n], -3) for n>0. - _Peter Luschny_, Nov 30 2014

%t a[0] = 1;

%t a[n_] := Sum[(n-m) Binomial[n+m-1, m] (-3)^m/n, {m, 0, n-1}];

%t Table[a[n], {n, 0, 22}] (* _Jean-Fran├žois Alcover_, Jul 30 2018 *)

%o (Sage)

%o def a(n):

%o if n == 0: return 1

%o return hypergeometric([1-n, n], [-n], -3).simplify()

%o [a(n) for n in range(24)] # _Peter Luschny_, Nov 30 2014

%o (PARI) a(n) = if (n==0, 1, sum(m=0, n-1, (n-m)*binomial(n-1+m, m)*(-3)^m/n)); \\ _Michel Marcus_, Jul 30 2018

%Y Cf. A064334, A000108.

%K sign,easy

%O 0,3

%A _Wolfdieter Lang_, Sep 21 2001

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Last modified February 15 20:49 EST 2019. Contains 320138 sequences. (Running on oeis4.)