A391302 - OEIS

%I #15 Dec 08 2025 08:58:51

%S 1,2,7,29,137,695,3695,20308,114436,657550,3837986,22692014,135619965,

%T 817991662,4972692185,30437169640,187422668362,1160229062798,

%U 7216344400527,45074402993754,282618594386957,1778172298429813,11223145772872875,71040444598820668

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

%H Vincenzo Librandi, <a href="/A391302/b391302.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..floor(n/3)} (-1)^k * (4*k+2) * binomial(3*n-5*k+2,n-3*k)/(3*n-5*k+2).

%t Table[ Sum[(-1)^k*(4*k+2)*Binomial[3*n-5*k+2,n-3*k]/(3*n-5*k+2),{k,0,Floor[n/3]}],{n,0,21}] (* _Vincenzo Librandi_, Dec 07 2025 *)

%o (PARI) a(n) = sum(k=0, n\3, (-1)^k*(4*k+2)*binomial(3*n-5*k+2, n-3*k)/(3*n-5*k+2));

%o (Magma) [&+[(-1)^k*(4*k+2)*Binomial(3*n-5*k+2, n-3*k)/(3*n-5*k+2): k in [0..Floor(n/3)]] : n in [0..30] ]; // _Vincenzo Librandi_, Dec 07 2025

%Y Cf. A391296, A391298, A391300.

%Y Cf. A001764, A391130.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 06 2025