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A364407 G.f. satisfies A(x) = 1 + x*(1 + 1/A(x)^3).

%I A364407 #19 Mar 03 2024 13:47:12
%S A364407 1,2,-6,42,-350,3234,-31878,328426,-3494142,38093442,-423344966,
%T A364407 4778162922,-54621614814,631114404258,-7358619459654,86472788963370,
%U A364407 -1023093071862526,12177054520248834,-145700860758056838,1751559565664348842,-21145576694586256734
%N A364407 G.f. satisfies A(x) = 1 + x*(1 + 1/A(x)^3).
%F A364407 G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A349310.
%F A364407 a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(n+3*k-2,n-1) for n > 0.
%F A364407 D-finite with recurrence 3*n*(52*n-187)*(3*n-1) *(3*n-2)*a(n) +(14392*n^4 -70190*n^3 +56951*n^2 +50237*n -49500)*a(n-1) +3*(-17252*n^4 +205959*n^3 -851664*n^2 +1432459*n -815652)*a(n-2) +18*(-472*n^4 +1294*n^3 +36359*n^2 -226731*n +361171)*a(n-3) -27*(n-5)*(404*n^3 -2235*n^2 -4058*n +26406)*a(n-4) -81*(n-5)*(n-6) *(8*n^2+358*n-1785)*a(n-5) +243*(n-5)*(n-6) *(n-7)*(4*n-31)*a(n-6)=0. - _R. J. Mathar_, Jul 25 2023
%p A364407 A364407 := proc(n)
%p A364407     if n = 0 then
%p A364407         1;
%p A364407     else
%p A364407         (-1)^(n-1)*add( binomial(n,k) * binomial(n+3*k-2,n-1),k=0..n)/n ;
%p A364407     end if;
%p A364407 end proc:
%p A364407 seq(A364407(n),n=0..70); # _R. J. Mathar_, Jul 25 2023
%t A364407 nmax = 20; A[_] = 1;
%t A364407 Do[A[x_] = 1 + x*(1 + 1/A[x]^3) + O[x]^(nmax+1) // Normal, {nmax+1}];
%t A364407 CoefficientList[A[x], x] (* _Jean-François Alcover_, Mar 03 2024 *)
%o A364407 (PARI) a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(n+3*k-2, n-1))/n);
%Y A364407 Cf. A364393, A364408, A364409.
%Y A364407 Cf. A349310, A364400.
%K A364407 sign
%O A364407 0,2
%A A364407 _Seiichi Manyama_, Jul 23 2023