A023423 - OEIS

%I #23 Feb 03 2025 09:35:33

%S 1,1,1,1,1,1,1,2,4,8,16,32,64,128,257,517,1042,2104,4256,8624,17504,

%T 35585,72455,147746,301706,616948,1263240,2589840,5316033,10924681,

%U 22475831,46290195

%N Generalized Catalan Numbers x^2*A(x)^2 -(1-x+x^2+x^3+x^4+x^5+x^6)*A(x) + 1 =0.

%H Seiichi Manyama, <a href="/A023423/b023423.txt">Table of n, a(n) for n = 0..3028</a>

%F G.f. A(x) satisfies: A(x) = (1 + x^2 * A(x)^2) / (1 - x + x^2 + x^3 + x^4 + x^5 + x^6). - _Ilya Gutkovskiy_, Jul 20 2021

%p A023423 := proc(n)

%p     option remember;

%p     if n <= 6 then

%p         1;

%p     else

%p         procname(n-1)+add(procname(k)*procname(n-2-k),k=5..n-2) ;

%p     end if;

%p end proc: # _R. J. Mathar_, Oct 10 2014

%t a[0]=1; a[n_]:= a[n]=a[n-1] + Sum[a[k]*a[n-2-k], {k,5,n-2}]; Table[a[n], {n,0,30}] (* modified by _G. C. Greubel_, Jan 01 2018 *)

%o (PARI) {a(n) = if(n==0,1, a(n-1) + sum(k=5,n-2, a(k)*a(n-k-2)))};

%o for(n=0,30, print1(a(n), ", ")) \\ _G. C. Greubel_, Jan 01 2018

%Y Cf. A000108, A001006, A004148, A004149, A023421, A023422.

%Y Sixth row of A064645.

%K nonn,easy

%O 0,8

%A _Olivier Gérard_