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%I #10 Jun 11 2024 00:10:56
%S 1,1,10,325,16750,1056250,74237500,5580578125,439118593750,
%T 35714849218750,2978473867187500,253316015488281250,
%U 21887247402929687500,1915840314586914062500,169529844641289062500000
%N Generalized Catalan numbers C(5,5; n).
%C See triangle A064879 with columns m built from C(m,m; n), m >= 0, also for Derrida et al. and Liggett references.
%H J. Abate, W. Whitt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Whitt/whitt6.html">Brownian Motion and the Generalized Catalan Numbers</a>, J. Int. Seq. 14 (2011) # 11.2.6, corollary 6.
%F a(n) = ((25^(n-1))/(n-1))*Sum_{m=0..n-2} (m+1)*(m+2)*binomial(2*(n-2)-m, n-2-m)*((1/5)^(m+1)), n >= 2, a(0) := 1 =: a(1).
%F G.f.: (1-9*x*c(25*x))/(1-5*x*c(25*x))^2 = c(25*x)*(9+16*c(25*x))/(1+4*c(25*x))^2 = (9*c(25*x)*(5*x)^2+8*(2+7*x))/(4+5*x)^2 with c(x)= A(x) g.f. of Catalan numbers A000108.
%F 4*(-n+1)*a(n) +5*(79*n-200)*a(n-1) +250*(2*n-3)*a(n-2)=0. - _R. J. Mathar_, Aug 09 2017
%Y Cf. A000108, A064342.
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_, Oct 12 2001
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