A391458 - OEIS

%I #12 Dec 21 2025 00:14:44

%S 1,2,9,51,335,2364,17511,134303,1057446,8498115,69424964,574844733,

%T 4813486921,40690941990,346798845213,2976661777689,25708160461067,

%U 223247734992435,1948116221860923,17073908081529946,150228705609723510,1326518390219586025

%N Expansion of g^2/(1 + x^3*g^3), where g = 1+x*g^4 is the g.f. of A002293.

%H Vincenzo Librandi, <a href="/A391458/b391458.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..floor(n/3)} (-1)^k * (3*k+2) * binomial(4*n-9*k+2,n-3*k)/(4*n-9*k+2).

%t Table[Sum[(-1)^k*(3*k+2)*Binomial[4*n-9*k+2,n-3*k]/(4*n-9*k+2),{k,0,Floor[n/3]}],{n,0,25}] (* _Vincenzo Librandi_, Dec 20 2025 *)

%o (PARI) a(n) = sum(k=0, n\3, (-1)^k*(3*k+2)*binomial(4*n-9*k+2, n-3*k)/(4*n-9*k+2));

%o (Magma) [&+[(-1)^k*(3*k+2)*Binomial(4*n-9*k+2, n-3*k)/(4*n-9*k+2): k in [0..Floor(n/3)]] : n in [0..25] ]; // _Vincenzo Librandi_, Dec 20 2025

%Y Cf. A002293, A391081.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 10 2025