A364758 - OEIS

%I #22 Dec 11 2024 08:45:34

%S 1,1,3,14,76,450,2818,18352,123028,843345,5884227,41650479,298352365,

%T 2158751879,15754446893,115830820439,857147952469,6379136387303,

%U 47715901304501,358529599468636,2704884469806606,20481615947325089,155605509972859999,1185779099027494848

%N G.f. A(x) satisfies A(x) = 1 + x*A(x)^4 / (1 + x*A(x)).

%H Seiichi Manyama, <a href="/A364758/b364758.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(4*n-3*k,n-1-k) for n > 0.

%F From _Seiichi Manyama_, Dec 11 2024: (Start)

%F G.f. A(x) satisfies A(x)^3 = 1 + x*A(x) + x*A(x)^5 + x*A(x)^6.

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

%F If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(s*k,n-k)/(t*k+u*(n-k)+r). (End)

%o (PARI) a(n) = if(n==0, 1, sum(k=0, n-1, (-1)^k*binomial(n, k)*binomial(4*n-3*k, n-1-k))/n);

%o (PARI) a(n, r=1, s=-1, t=4, u=1) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r)); \\ _Seiichi Manyama_, Dec 11 2024

%Y Cf. A090192, A106228, A364759, A378919.

%Y Cf. A001764, A219537, A364865, A365224.

%Y Cf. A300048, A364747, A378889.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Aug 05 2023