A378670 - OEIS

%I #17 Dec 04 2024 08:39:29

%S 1,2,11,78,627,5432,49464,466726,4522871,44747874,450127999,

%T 4589821576,47333631828,492836382192,5173697858508,54700317431958,

%U 581946708333055,6225343630256678,66921440314606905,722546760572660030,7832054418695360555,85198490262065775840

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

%F G.f.: exp( 2/3 * Sum_{k>=1} A378612(k) * x^k/k ).

%F G.f.: B(x)^2 where B(x) is the g.f. of A243659.

%F a(n) = 2 * Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(n,k) * binomial(3*n+k+2,n)/(3*n+k+2).

%F a(n) = 2 * Sum_{k=0..n} binomial(3*n+k+2,k) * binomial(n-1,n-k)/(3*n+k+2).

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

%o (PARI) a(n) = 2*sum(k=0, n, 2^k*(-1)^(n-k)*binomial(n, k)*binomial(3*n+k+2, n)/(3*n+k+2));

%o (PARI) a(n, r=2, s=1, t=4, u=3) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

%Y Cf. A243659, A369012.

%Y Cf. A010683, A211789, A378668.

%Y Cf. A378612.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 02 2024