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A376486 G.f. satisfies A(x) = 1 / (1 - x^3*A(x)^3 * (1 + x)).

%I A376486 #12 Sep 26 2024 04:09:30
%S A376486 1,0,0,1,1,0,4,8,4,22,66,66,162,560,840,1529,4985,9690,16774,47349,
%T A376486 107229,195500,483000,1172724,2311516,5249556,12910716,27299992,
%U A376486 59765400,144602352,321554224,700449496,1654540452,3789265198,8344514618,19327204006
%N A376486 G.f. satisfies A(x) = 1 / (1 - x^3*A(x)^3 * (1 + x)).
%H A376486 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A376486 G.f.: (1/x) * Series_Reversion( x*(1-x^3)/(1+x^4) ).
%F A376486 a(n) = Sum_{k=0..floor(n/3)} binomial(4*k,k) * binomial(k,n-3*k) / (3*k+1).
%F A376486 D-finite with recurrence 243*n*(n-1)*(n+1)*a(n) +81*n*(n-1)*(14*n-31)*a(n-1) +27*(73*n-240)*(n-1)*(n-2)*a(n-2) +36*(-22*n^3-99*n^2+731*n-870)*a(n-3) +48*(-263*n^3+1908*n^2-4449*n+3290)*a(n-4) -128*(n-4)*(230*n^2-1429*n+2205)*a(n-5) -768*(n-5)*(43*n-168)*(n-4)*a(n-6) -9216*(n-5)*(n-6)*(2*n-9)*a(n-7) -4096*(n-5)*(n-6)*(n-7)*a(n-8)=0. - _R. J. Mathar_, Sep 26 2024
%p A376486 A376486 := proc(n)
%p A376486     add(binomial(4*k,k)*binomial(k,n-3*k)/(3*k+1),k=0..floor(n/3)) ;
%p A376486 end proc:
%p A376486 seq(A376486(n),n=0..70) ;
%p A376486 # _R. J. Mathar_, Sep 26 2024
%o A376486 (PARI) a(n) = sum(k=0, n\3, binomial(4*k, k)*binomial(k, n-3*k)/(3*k+1));
%Y A376486 Cf. A005043, A052709, A376487.
%K A376486 nonn
%O A376486 0,7
%A A376486 _Seiichi Manyama_, Sep 25 2024