A371542 - OEIS

%I #16 Dec 10 2024 05:55:12

%S 1,3,9,34,141,621,2849,13467,65127,320686,1602294,8103087,41397186,

%T 213331026,1107604764,5788249329,30422897664,160717169622,

%U 852894534042,4544635033164,24305345593290,130423538829518,701994030831654,3788979493701069,20503322609731348

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

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

%F From _Seiichi Manyama_, Dec 10 2024: (Start)

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

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

%F G.f.: A(x) = (1 + B(x))^3 where B(x) is the g.f. of A127897. (End)

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

%Y Cf. A127897, A370720, A371494.

%Y Cf. A371495, A371538.

%Y Cf. A371516.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 26 2024