A395704 - OEIS

%I #13 May 04 2026 11:02:32

%S 1,14,144,1308,11105,90360,714084,5524152,42046371,315964470,

%T 2349970656,17329573296,126885222828,923410872944,6685074723420,

%U 48177037031280,345810084667287,2473424154340794,17635590347130816,125387174972644140,889218363195578241

%N Expansion of g^4*(2*g-1)/(3-2*g)^4, where g = 1+x*g^3 is the g.f. of A001764.

%F a(n) = (n+1) * A036829(n+1).

%F G.f.: (Sum_{k>=0} (4*k+1)/(3*k+1) * binomial(3*k+1,k) * x^k) * (Sum_{k>=0} binomial(3*k,k) * x^k)^3.

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

%F a(n) = (n+1) * Sum_{k=0..n} binomial(3*k+1+l,k) * binomial(3*n-3*k-l,n-k) for every real number l.

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

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

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

%Y Cf. A395705, A395706, A395707.

%Y Cf. A001764, A036829, A378504.

%K nonn

%O 0,2

%A _Seiichi Manyama_, May 04 2026