A379081 - OEIS

%I #13 Dec 15 2024 06:42:35

%S 1,3,15,94,657,4905,38299,308928,2554092,21528728,184318944,

%T 1598427531,14011401996,123946608699,1105090991634,9920335032821,

%U 89589290332200,813367589142888,7419376746340780,67965042988027335,624971955439306953,5766825797557702751,53380176096582823851

%N Expansion of (1/x) * Series_Reversion( x * (1/(1 + x) - x^2)^3 ).

%F G.f. A(x) satisfies:

%F (1) A(x) = exp( Sum_{k>=1} A379086(k) * x^k/k ).

%F (2) A(x) = ( (1 + x*A(x)) * (1 + x^2*A(x)^(7/3)) )^3.

%F (3) A(x) = B(x)^3 where B(x) is the g.f. of A379088.

%F a(n) = (1/(n+1)) * [x^n] 1/( 1/(1 + x) - x^2 )^(3*(n+1)).

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

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

%Y Cf. A181734, A379080.

%Y Cf. A379086, A379088.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 15 2024