A378891 - OEIS

%I #11 Dec 10 2024 06:15:19

%S 1,3,18,142,1278,12429,127223,1350456,14729628,164079982,1858781652,

%T 21348787587,248021665720,2909439099543,34413536180688,

%U 409984974779725,4915119769384221,59252402698999209,717819918438472134,8734481867945979183,106703642464149880248

%N G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^2/(1 + x*A(x)) )^3.

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

%F G.f. A(x) satisfies A(x) = 1 + x * A(x) * (1 + A(x)^(4/3) + A(x)^(5/3)).

%F G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A378892.

%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).

%o (PARI) a(n, r=3, s=-1, t=6, u=3) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));

%Y Cf. A371542, A378889, A378890.

%Y Cf. A378892.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 10 2024