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 A064092 Generalized Catalan numbers C(9; n). 5

%I

%S 1,1,10,181,4078,102826,2777212,78571837,2298558934,68964092542,

%T 2110472708140,65620725560578,2067160250751436,65833929303952564,

%U 2116166898185821792,68565914052628406221,2237022199842087256678

%N Generalized Catalan numbers C(9; n).

%C a(n+1)= Y_{n}(n+1)= Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=9, beta =1 (or alpha=1, beta=9).

%F G.f.: (1+9*x*c(9*x)/8)/(1+x/8) = 1/(1-x*c(9*x)) with c(x) g.f. of Catalan numbers A000108.

%F a(n)= sum((n-m)*binomial(n-1+m, m)*(9^m)/n, m=0..n-1) = ((-1/8)^n)*(1-9*sum(C(k)*(-72)^k, k=0..n-1)), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).

%F a(n) = Sum{ k= 0...n, A059365(n, k)*9^(n-k) } . - _Philippe Deléham_, Jan 19 2004

%F Conjecture: 8*n*a(n) +(-287*n+432)*a(n-1) +18*(-2*n+3)*a(n-2)=0. - _R. J. Mathar_, Jun 07 2013

%t a[0] = 1; a[n_] := Sum[(n - m)*Binomial[n - 1 + m, m]*9^m/n, {m, 0, n - 1}]; Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Jul 09 2013 *)

%o (PARI) a(n)=if(n<0,0,polcoeff(serreverse((x-8*x^2)/(1+x)^2+O(x^(n+1))),n)) /* _Ralf Stephan+ */

%Y A064091 (C(8, n)).

%K nonn,easy

%O 0,3

%A _Wolfdieter Lang_, Sep 13 2001

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Last modified February 21 06:40 EST 2019. Contains 320371 sequences. (Running on oeis4.)